Transcript Document

GRAPHS
CHAPTER 1
Contents
1.
2.
3.
4.
Families of simple graphs
Metric space.
Valence, Grith and Cages
Isomorhpism, Matrices and Graph
Invariants
5. Connectivity in Graphs
6. Subgraphs
7. Basic Operations of graphs
8. Advanced Operations on Graphs
9. Variations of Graphs
10. Graph Products
1.
2.
3.
4.
5.
6.
7.
Factors and Factorizations
Planar Graphs
Graphs from Polyhedra
Metric Space - Revisited
Representations of Graphs
Edge-Colorings and Snarks
Vertex Colorings
1. Families of Simple Graphs
Graphs
a
1
2
c
d
b
e
3
4
• Simple graph G=(V,E)
• V = V(G) ={1,2,3,4} –
vertices
• E = E(G) = {a,b,c,d,e}
– edges
• Edge a has endvertices 1 and 2.
Vertices 1 and 2 are
adjacent: 1 ~ 2.
Simple Graph
• Definition: Graph X is composed of a set of
vertices V(X) endowed with an irreflexive
symmetric relation ~ (adjacency). An
unordered pair of adjacent vertices uv = vu
forms an edge. The set of edges is denoted
by E(X). Sometimes we write X = (V,E) or
X(V,E).
Cycle Cn on n vertices.
1
2
3
4
C4
V – vertices of a regular
n-gon
E – edges
• |V|=n
• |E|=n
Small Cycles
C3
C5
C4
C6
• Some cycles as drawn by
VEGA.
• It makes sense to define
cylces C1 (a loop) and C2
(parallel edges), that are
NOT simple.
C1
C2
Path Pn on n vertices.
1
P4
V – vertices of polygonal
line.
E – segments.
The endpoints of the
polygonal line are called
the endpoints of the path.
For instance, 1 and 4 are the
endpoints of the path on
the left.
• |V|=n
• |E|=n-1
2
3
1
2
3
4
4
Complete graph on n vertices Kn.
1
2
3
4
K4
V – vertices of a regular
n-gon
E – n-gon edges and
diagonals.
• |V|=n
• |E|=n(n-1)/2
Complete Bipartite Graph on
n+m vertices Kn,m.
1
2
V = U1  U2 , U1 Å U2 = ;
|U1| = m, |U2 | = n.
E = U1  U2
3
4
K2,2
• |V|=n + m
• |E|=n m
The Petersen Graph and its
Generalizations G(n,k)
• Petersen graph G(5,2) is
an example of a
generalized Petersen
graph G(n,k).
• V(G(n,k)) consists of
• ui, vi, i = 1,2, ..., n.
Edges:
• ui ~ ui+1
• ui ~ vi
• vi ~ vi+k
(Warning! Addition mod n)
Examples of Generalized
Petersen graphs
• G(10,2) Dodecahedron
• G(10,3) Desargues
graph.
• G(8,3) Möbius-Kantor
graph.
• G(6,2) Dürer graph.
2. Metric space
Metric Space
• Space V, with mapping d (distance):
• d:V  V  R with the following properties:
•
•
•
•
d(u,v) ¸ 0, d(u,v) = 0, iff u = v.
d(u,v) = d(v,u)
d(u,v) · d(u,w) + d(w,v)
is called a metric space with distance d.
Example: Hamming Distance
{0,1}n is a metric space if the distance
between u and v is the number of
components in which the two vectors
differ.
–
–
E.g. d([0,0,0,1,0,1],[1,1,0,1,1,1]) = 3.
d is called the Hamming distance.
Hypercube Qn.
Q1
Q4
Q2
Q3
Q5
• Hypercube of
dimension d is the
graph Qn, with:
• V(Qn) = {0,1}n.
• u ~ v, if d(u,v) = 1.
• |V(Qn)| = 2n
• |E(Qn )|= n 2n-1
3. Valence, Girth and Cages
Vertex Valence
•
•
•
•
a
1
2
c
d
b
e
3
4
•
•
•
•
G = (V,E)
V(G) ={1,2,3,4}
E(G) = {a,b,c,d,e}
the number of edges incident
with vertex v is called the
valence or degree of v: deg(v).
deg(1) = deg(4) = 3, deg(2) =
deg(3) = 2.
a vertex of valence 1 is called a
leaf, a vertrex of valence 0 is
isolated.
d(G) – minimal valence.
D(G) – maximal valence.
Regular Graphs
Graph G is regular (of
valence k), if d(G) =
D(G) = k.
Examples:
• Regular graphs: Kn, Cn,
Kn,n
• Nonregular graphs: Pn, n >
2, Kn,m, n  m.
1-valent and 2-valent graphs
have simple structure.
Trivalent graphs have a
special name: cubic
graphs. (See example on
the left)
Girth
• Girth g(G) of graph G is the number of
vertices of the shortest cycle in G. If G has
no cycles, its girth is infinite.
Cages
•
Graph G is a g-cage, if the following
holds:
1. Trivalent
2. Has girth g
3. Has the least number of vertices among the
graphs satisfying 1 and 2.
Exercises 3
•
•
•
•
N1. Determine the 3-cage.
N2. Determine the 4-cage.
N3. Determine the 5-cage.
N4. Determine the 6-cage.
4. Isomorphism, Matrices and
Graph Invariants
Incidence Matrix M(G).
• To G=(V,E) we associate a rectangular
matrix M=M(G) with |V| rows and |E|
columns:
Mv,e =
{
1 ...
v is the endpoint of e
0 ...
otherwise
Incidence Matrix - Example
• G=(V,E)
• VG ={1,2,3,4}
• EG = {a,b,c,d,e}
a
1
2
c
d
b
e
3
4
1
MG =
2
3
4
a
1
1
0
0
b
0
1
0
1
c
1
0
0
1
d
1
0
1
0
e
0
0
1
1
Handshaking Lemma
• In each graph G=(V,E) :
• 2 |E(G)| = Sv 2 V(G) deg(v),
• The proof uses the so-called bookkeeper’s
rule in the incidence matrix of graph G.
Graph Invariant
• It is well-known that we associate numbers to
mathematical objects in various ways. For instance:
Determinant is assicated with a matrix, degree is
associated with a polynomial, dimension is associated
with a space, length is associated with a vector, etc.
• There are several numbers that can be associated with a
graph. Such a number is usually called graph invariant.
One may argue that the main topic of graph theory is the
study of graph invariants.
• In addition to numbers other objects may be graph
invariants.
Isomorphisms and Graph Invariants
An isomorphism s(G) = H is a bijective
mapping:
• s: V(G) ! V(H).
that preserves adjacency:
• u ~ v if and only if s(u)~s(v).
A graph invariant is a property, (usually a
number), that is preserved under an
isomorphism.
Isomorphism - Exercises
A
C
B
D
•
•
N1. Determine an
isomorphism
between graphs A
and B.
N2. Determine an
isomorphism
between graphs C
and D.
Adjacency Matrix A(G).
• To each graph G=(V,E) with V={1,2,3,...,n}
we can associate the adjacency matrix
A=A(G) as follows:
Ai,j =
{
1 ...
i~j
0 ...
sicer
Adjacency Matrix - Example
• G=(V,E)
• VG ={1,2,3,4}
• EG = {a,b,c,d,e}
a
1
2
c
d
b
e
3
4
0
1
AG =
1
1
1
0
0
1
1
0
0
1
1
1
1
0
Adjacency Matrix is Not an
Invariant
• The adajcency matrix is not an invariant. It
depends on the numbering of vertices.
• The incidence matrix is not an invariant. It
depends on the numbering of the vertices
and ordering of the edges.
Some Graph Invariants
•
•
•
•
|V(G)| = number of vertices
|E(G)| = number of edges
d(G) = minimal valence.
D(G) = maximal valence
Invariants - Example
a
1
2
c
d
b
e
3
4
•
•
•
•
|V(G)| = 4
|E(G)| = 5
d(G) = 2
D(G) = 3
5. Connectivity in Graphs
Disjoint Union of Sets
• Let A and B be sets. By A t B we denote the
disjoiont union of A and B. If A Å B = ;,
then A t B is simply the union of the two
sets. Otherwise we defne formally A t B =
A £ {0} [ B £ {1}.
Disjoint Union of Graphs
• Let G’ and G” be graphs. By G’ t G” we
denote the disjoiont union of graphs G’ and
G”. This means
• V(G’ t G”) := V(G’) t V(G”) and
• E(G’ t G”) := E(G’) t E(G”).
The Empty Graph
• Empty graph f = (f,f) has no vertices and
no edges.
Connectivity in Graphs - Theory
• Graph G is connected, if and only if it
cannot be written as a disjoint union of two
non-empty graphs.
Connectivity of Graphs - Practice
• Graph is connected, if we grab and shake
the “model” made of balls and strings, and
nothing falls down to earth. (No knotting of
strings is permitted!)
Equivalence Relation @.
• Let G be a graph. On V(G) define @ as
follows: For any u,v 2 V(G) let u @ v, if and
only if there exists a subgraph, isomorphic
to a path with endpoints u and v.
• Proposition. @ is an equivalence relation on
V(G).
• Proof. Obviously reflexive and symmetric.
Proof of transitivity – Homework.
Path Connectivity of Graphs
• G is connected by paths, if the
equivalence relation @ has a single
equivalence class.
Trees
• A tree is a connected graph with no cycles
• There are several characterizations of tree,
such as:
• A tree is a connected graph with n vertices and n-1
edges.
• A tree is a connected graph that is no longer
connected after removal of any edge.
• A tree is connected and cycle free.
Bipartite Graphs
• A graph is bipartite, if
the vertex set can be
partitioned into two
bipartitions, say G and
R, such that each edge
has one endpoint in G
and the other in R.
• The graph on the left
is biparitite.
Exercises 5
• N1: Show that each Km,n. is bipartite.
• N2: Show that each Qn is bipartite.
• N3(*): Show that a graph is bipartite if and only if
it has no odd cycles.
• N4: Which generalized Petersen graphs G(n,k) are
bipartite?
• N5: Prove that each tree is a bipartite graph.
• N6: Prove that X is bipartite, if and only if each of
its components is bipartite.
Homework 5
• H1: Prove that the relation @ is transitive.
• H2: Prove that for finite graphs the notions
of connectedness and path connectedness
coincide.
6. Subgraphs
Subgraphs
• Graph H=(U,F) is a subgraph of graph
G=(V,E), if U µ V and F µ E.
• Warning! It is important that (U,F) is
indeed a graph! Each edge from F must
have both of its endpoints in U.
Subgraphs - Example
a
1
2
c
d
b
e
3
4
• G=(V,E)
• VG ={1,2,3,4}
• EG = {a,b,c,d,e}
Let: U = {1,2,3}, W =
{2,3,4}, F = {b}, P =
{a,d}. Then (U,P) and
(W,F) are subgraphs
while (U,F) and (W,P)
are not.
Subgraph Types
•
•
•
•
•
Open subgraph
Induced subgraph
Spanning subgraph
Isometric subgraph
Convex subgraph
Open Subgraph
• Subgraph H=(U,F) of graph G=(V,E) is
open, if each ede e 2 E has either both
endpoints in U, or none.
Trivial Subgraph
• Subgraph H is trivial, if either H = f, or H
= G.
Connected Component
• A minimal nontrivial open subgraph is
called a connected component of G. By
W(G) we denote the number of connected
components of graph G.
Distance in a Connected Graph
• Each connected graph G gives rise to a
metric space (V,dG) for dG(u,v) being the
length of a shortest path in G, from u to v.
Distance Partition
• For a given graph G and a given vertex v we may
define the k-th link: Vk := {u 2 V(G)| d(v,u) = k}.
• This defines a partition V = {V0,V1,...,Ve} , Vk  ;
of the vertex set V(G) = V0 t V1 t ... t Ve. The
number e is called the excentricity of vertex v.
The maximum excentricity is called the diameter
of graph.
• This partition is called the distance partition of G
with respect to v.
• Clearly, V0 = {v}.
k-connectedness
• Graph G with |V(G)| > k is k-connected, if the
removal of any set S with |S| < k leaves a
connected graph.
• Connectivity k(G) of graph G is the largest k,
such that G is still k-connected.
• Vertex v of graph G is a cut-vertex, if W(G – v)
> W(G ).
• A connected graph with no cut-vertex is called a
block.
2-connectedness
• Theorem: The following claims are
equivalent:
– Graph G is 2-connected,
– Graph G is a block,
– Any pair of vertices belongs to a common
cycle.
Menger’s Theorem
• Two paths in a graph with a common pair of
end-vertices are internally disjoint, if they
have no other vertex in common.
• Theorem: Graph is k-connected, if and only
if there are k pair-wise internally disjoint
paths between any two of its vertices.
Spanning Subgraph
• If H=(U,F) is a subgraph of G(V,E) and U =
V, then H is called a spanning subgraph of
G.
Spanning Paths and Cycles
• A spanning subgraph is also called a factor.
• A spanning path in a graph is also called a
hamilton path.
• A spanning cycle in a graph is also called a
hamilton cycle.
Spanning Trees
• Each connected graph has a spanning tree.
• For finite graphs the proof is not hard: A
connected graph which is not a tree must contain a
cycle. Removing a single edge from a cycle does
not destroy connectivity. We may continue to
remove edges from cycles until there is no cycle
left, i.e. we obtain a spanning tree.
• For infinite graphs this fact is equivalent to the
axiom of choice.
How many spanning trees does
the complete graph have?
• K3 has three spanning
trees!
• Let t(G) denote the
number of spanning
trees in G.
• Theorem: t(Kn) = nn-2
• Proof: Prüfer code!
Induced Subgraph
• Graph H is an induced subgraph
of graph G, if H is obtained
from G by removing the
vertices from V(G)-V(H).
• An induced subgraph of G is
determined by its vertrex set U
µ V(G). If we want to
distinguish the graph from its
vertex set we denote the former
by <U> or, if we want to refer
to the original graph by G|U.
• Example: P5 is an induced
subgraph of C6.
Isometric Subgraph
• H=(U,F) is an isometric subgraph of graph
G=(V,E), if the distances are preserved:
• For each u,v 2 U: dH(u,v) = dG(u,v).
Interval IG(u,v)
• Let u, v 2 V(G) belong to the same
connected component of G. By IG(u,v) we
denote the interval with endpoints u and v.
• IG(u,v) is the graph, induced on the set of
vertices belonging to some shortest path
from u to v.
• If there is no danger of confusion we can
simplify notation: I(u,v).
Convex Subgraph
• Graph H is a convex subgraph of G, if for
every pair of vertices u and v from V(H)
that belong to the same connected
component of G, the interval IG(u,v) is a
subgraph of H.
Exercises 6-1
• N1. Prove that G is connected if and only if it has no
nontrivial open subgraphs.
• N2. Show that if G has a hamilton cycle it also contains a
hamilton path.
• N3. Show that every graph that has a hamilton path is
connected.
• N4. Construct a graph on 10 vertices that has no hamilton
path.
• N5. Construct a graph on 10 vertices that has no hamiloton
cycle but has a hamilton path.
• N6: Construct a graph on 10 vertices that has a hamilton
cycle.
Exercises 6-2
• N7. Prove the following: In a connected graph G there
exsists at least one distance partition such that each k-link
Vk is an independent set if and only if G is bipartite.
• N8. Let G and H be graphs. We say, that G is locally H if
and only if for each vertex v 2 V(G) the first link <V1(v)>
is isomorphic to H. Find a graph that is locally P3.
• N9. Prove that K2,2,2 is locally C4.
• N10. Determine all graphs with diameter 1.
• N11. Use the result of N7 to show that if one distance
partion has independent k-links then all of them have
independent k-links.
• N12. Use N11 to design an algorithm that will find a
bipartition of a bipartite connected graph.
Exercises 6-3
• N13. Prove that each convex subgraph is an isometric subgraph.
• N14. Prove that each isometric subgraph is an induced
subgraph.
• N15. Prove that each connected component is a convex
subgraph.
• N16. Prove that the intersection of two induced subgraphs is an
induced subgraph.
• N17. Prove that the intersection of two convex subgraphs is a
convex subgraph.
• N18. Determine all intervals of the cube Q3.
Exercises 6-4
7
6
8
1
2
• N19. For H µ G define the convex
5
closure cvx(H) of H in G. Compute
cvx(Pk) in Cn.
• N20. Prove that each interval I(a,b) is
a subgraph of cvx(a,b).
• N21. Determine all intervals in the
graph G on the left. Find two vertices
a and b of G that have I(a,b) 
4
cvx(a,b).
• N22. Prove that althouth the subgraph
induced by any shortest path in G is
isometric, there are intervals that are
not isometric subgraphs.
• N23. Prove that each interval in a tree
3
is a path.
• N24. Characterize graphs, with the
property that each interval is a path.
Homework 6
• H1. Let C be the shortest cycle in graph G.
Show that C is an induced subgraph of G.
• H2. Determine all non-isomorphic intervals
in Q4.
• H3. Find an isometric subgraph of Q3 that is
not convex.
7. Basic Operations on Graphs
Basic Operations on Graphs
•
•
•
•
•
•
Deletion of edges
Deletion of vertices
Addition of edges
Union
Complement
Join
Deletion of Edges
• If G = (V,E) is a graph and e 2 E one of tis
edges, then G - e := (V,E – {e}) is a
subgraph of G. In such a case we say that
G-e is obtained from G by deletion of edge
e.
Deletion of Vertices
• Let x 2 V(G) be a vertex of graph G, then
G - x is the subgraph obtained from G by
removal of x grom V(G) and removal of all
edges from E(G) having x as an endpoint.
G – x is obtained from G by deletion of
vertex x.
Edge Addition
• Let G be a graph and (u,v) a pair of nonadjacent vertices. Let e = uv denot the new
edge between u and v. By G’ = G + uv = G
+ e we denote the graph obtained from G by
addition of edge e. In other words:
• V(G’) : = V(G),
• E(G’) : = E(G) [ {e}.
Graph Union Revisited
• If G and H are graphs we
denote by G t H their disjoint
union.
• Instead of G t G we write 2G.
• Generalization to nG, for an
arbitrary positive integer n:
– 0G := ;.
– (n+1)G := nG t G
• Example:
• Top row : C6 t K9
• Bottom row: 2K3.
Graph Complement
• The graph
complement Gc of a
simple graph G has
V(Gc) := V(G), but
two vertices u and v
are adjacent in Gc if
and only if they are
not adjacent in G.
• For instance C4c is
isomorphic to 2K2.
Graph Difference
• If H is a spanning
subgraph of G we may
define graph
difference G \H as
follows:
• V(G\H) := V(G).
• E(G\H) := E(G)\E(H).
G
H
G\H
Bipartite Complement
X
Xb
• For a bipartite graph X
(with a given
biparitition) one can
define a bipartite
complement Xb. This
is the graph difference
of Km,n and X: Xb =
Km,n \ X.
Empty Graph Revisited.
• The word “empty graph” is used in two
meanings.
• First Meaning: ;. No vertices, no edges.
• Second Meaning: En := Knc.= nK1. There are
n vertices, no edges.
• E0 = ; = 0. G will be called the void graph
or zero graph.
Graph Join
• Join of graphs G and H is denoted by G*H
and defined as follows:
• G*H := (Gc t Hc )c
• In particular, this means that Km,n is a join
of two empty graphs En and Em.
Exercises 7
• N1. Show that for any set F µ E(G) the graph G-F
is well-defined.
• N2. Show that for any set X µ V(G) the graph GX is well-defined.
• N3. Show that for any set X µ V(G) and any set
F µ E(G) the graph G-X-F is well-defined.
• N4. Prove that H is a subgraph of G if and only if
H is obtained from G by a succession of vertex
and edge deletion.
8. Advanced Operations on
Graphs
Cone and Suspension
• The join of G and K1 is called the cone over
G and is denoted by Cone(G) = G*K1.
• The join G*(2K1 ) is called suspension.
Examples
• Any complete multipartite graph is a join of empty
graphs.
• The cone Cone(Cn) is called a pyramid or wheel
Wn.
• The octahedral graph is the suspension over C4.
It can be written in the form:
– O3 = (2K1)*(2K1)*(2K1).
• Construction can be generalized to:
– On = (2K1)*(2K1)* ...*(2K1)
Subdivision
• Let e 2 E(G) be an edge of G. Let S(G,e) denote the graph
obtained from G by replacing the edge e by a path of
length 2 passing through a new vertex. Such an operation
is called subdivision of the edge e..
• Let F be a subset of E(G), then S(G,F) denotes the graph
obtained from the subdivision of each edge of F. In the
case F = E, we drop the second argument and S(G)
denotes the subdivision graph of G.
• Graph H is a general subdivision of graph G, if H is
obtained from G by a sequence of edge subdivisions.
Graph Homeomorphism
• Graphs G and H are homeomorphic, if they have a
common subdivision.
• Graph G is topologically contained in a graph K, if there
exists a subgraph H of K, that is homeomorphic to G.
Matching
• Edges with no common endvertex are called
independent. A set of pairwise independent
edges is called a matching.
Maximal Matching
• A matching that cannot be augmented by
adding new edges is called a maximal
matching.
Perfect Matching
• Proposition: Let M be a matching of a
graph G on n vertices. Then |M| · n/2.
• A matching M with |M| = n/2 is called a
perfect matching.
Abstract Simplicial Complex
a
d
b
c
e
h
f
g
• K µ P(S) is an
abstract simplicial
complex if for each s
2 K and each t µ s it
follows that t 2 K.
• On the left:
• K = {;, a, b, c, d, e, f,
g, h, ab, ad, abd, bc,
be, bce, bd, ce, df, dg,
de, eh}
Line Graph L(G)
• Two edges with a common end-vertex are
incident. Incidence is a binary relation on
the edge set E(G).
• Line graph L(G) has the vertex set E(G),
while the edges of L(G) are determined by
the incidence of edges in G.
Examples
• The top row depicts
the Heawood graph
and its fourvalent
linegraph.
• The bottom row
depicts the Petersen
graph and its line
graph.
Exercises 8-1
•
•
•
N1: Prove that for any graph X at least one of the
graphs X and Xc is connected.
N2: Decribe two graphs G and H, so that H is
isomorphic to an induced subgraph, and also to a
non-induced subgraph of G.
N3: The graph of our original example is
isomorphic to K4-e.
1.
2.
3.
4.
How many subgraphs does K4-e have?
How many subgraphs of K4-e are non-isomorphic ?
How many induced subgraphs does K4-e have ?
How many of the induced subgraphs of K4-e are nonisomorphic?
Exercises 8-2
• N4. Determine the number of vertices and egdes
of the generalized octahedral graph On.
• N5. Let V = {-1,1}n. Define a graph Gn, whose
vertex set is V and two vertices are adjacent if and
only if d(u,v)2 < n. Prove that Gn is isomorphic to
On.
• N6. Explore the relationship between graphs G1 =
G * (2K1) and G2 = ((G*K1)*K1).
• N7. True or False? The Cone(H) is convex in
Cone(G) if H is convex in G.
• N8. Show that G\H is a spanning subgraph of G.
Exercises 8-3
• N9: Prove that S(G) is bipartite for any simple G.
• N10: Find a general subdivision of K3,3 in G(5,2).
• N11: Given a graph X with n vertices and m
edges. Determine the number of verticexs and the
number of edges of S(X)?
• N12: Graph G is topologically almost contained
in graph K, if there exists a subgraph H of K, that
is a general subdivision of G. Prove that
topological containment implies topological
almost containment and find a counterexample for
the converse.
Exercises 8-4
• N13. Prove that |M| · n/2 for any matching.
• N14. Prove that if a graph has a perfect matching
then it has an even number of vertices.
• N15. Prove that a parfect matching is a maximal
matching. Give an example of a maximal
matching that is not perfect.
• N16. Show that the family of matchings defines an
abstract simplicial complex on E(G).
Exercises 8-5
• N17: Find a perfect matchning in the
Petersen graph G(5,2).
• N18: Find a cubic graph with no perfect
matching.
Exercises 8-6
• N19: Show that the line graph of a regular graph is regular.
• N20: What is the number of vertices and the number of
edges of L(G), given the number of vertices n and the
number of edges m in G?
• N21: The trunacation T(G) of G is defined by T(G) :=
L(S(G)). What is the number of vertices (edges) in T(G)?
• N22: Draw T(K4) and T(Q3).
• N23: Define operations L(G) and T(G) only partially, for
some set of vertices or edges, similarly to the definition of
S(G,F). For instance, for bipartite graphs we may define
operations with respect to only one bipartition set.
Homework 8
• H1. Prove that C8 is isomorphic to its bipartite
complement C8b.
• H2. Determine all paths Pn that are isomorphic to their
bipartite complements Pnb.
• H3. Draw the suspension over C5.
• H4. A graph which is isomorphic to its complement is
called self-complementary. Prove that there exists no selfcomplementary graph on (n+2) vertices, if there exist selfcomplementary graphs on n and on (n+1)-vertices.
• H5. Draw all self-complementary paths and all selfcomplementary cycles.
9. Variations of Graphs
Variations of Graphs
• Our main topic are simple finite graphs. We
have to be aware of other similar creatures.
•
•
•
•
•
•
Infinite graphs
Digraphs
General graphs (multigraphs)
Pregraphs
Rooted graphs
...
Infinite Graphs
• An infinite graph may have an infinite set of
vertices or edges. Countable graphs have both sets
coutable (or finite)
• Among countable graphs most tractable are
locally finite graphs. (Each vertex has finite
valence).
• Even more restricted are bouded valence graphs,
where D(G) is finite, in particular regular ones.
Examples
• Examples of infinite graphs:
• Each metric space (X,d) determines a unit distance graph:
» V(X) := X
» x ~ y , d(x,y) = 1.
•
•
•
•
•
•
Ray or infinite path P1
Double ray or infinite cycle C1
Infinite k-way tree T(1,k).
Infinite square lattice Q1.
Infinite triangular lattice T1.
Infinite hexagonal lattice H1.
Digraphs
• In case of directed graphs (digraphs) there
are no undirected edges e = uv, but there are
directed edges or arcs: a = (u,v).
u
u
e = uv
a = (u,v)
v
v
Digraphs from Binary Relations
• Let R µ V £ V be a binary realtion on V. R
defines a digraph in a natural way.
Loops, parallel edges,
multigraphs
• A multigraph (general graph) is more
general than a simple graph, since it may
have parallel edges (between vertices u and
v we may have two edges e = uv and f = uv)
or loops. A loop z = uu is an edge whose
endpoints coincide.
u
v
C2, parallel edges
u
C1, loop
Dipoles, bouquets of circles and
more
Q3
B4
The handcuff graph G(1,1)
• Among general graphs
there are several
interesting families.
• A dipole qn has two
vertices and n parallel
edges between them.
• A bouqet of circles Bn has
one vertex with n loops.
• The handcuff graph
G(1,1) is a graph with two
adjacent vertices, each one
having a loop.
Pregraphs
• If we want to distinguish a loop from a halfedge we need the notion of a pregraph.
Definition of pregraph
• Pregraph X = (V,S,i,r) consists of
•
•
•
•
vertex set V,
set of semi-edges S,
mapping intial vertex, i: S ! V
involution reverse r: S ! S.
• Since r is an involution, we have r2 = 1. The orbits
of r of length 2 are edges, while orbits of length 1
are half-edges (semi-edges).
• A pregraph without half-edges is a general graph.
Rooted Graphs
• Let X be a graph and let r 2 V(X) be a selected
vertex. A pair (X, r) is called a rooted graph
and x is called the root.
• The idea of root can be generalized to a set of
vertices or even a subgraph Y µ X. In some
computer systems the root graph Y corresponds to
the selected part of X.
• The idea can be generalized to “graduated”
graphs:
• Y0 = r µ Y1 µ ... µ Yk = X.
Edge Contraction
• Let e = uv be an edge of G. By G/e we
denote the graph obtained from G by
contraction of e. This means we identify
vertices u and v, remove the loop and
identify possible extra parallel edges.
Exercises 9-1
• N1: Following the definion of a category, define a
digraph as a quadruple (V,A,i,t), where i and t are
mappings, that assign to each arc a its intial vertex
i(a) and terminal vertex t(a). Explain the procedure
that starts with a binary relation R and obtains a
digraph (V,A,i,t).
• N2: Define homomorphisms and isomorphisms of
digraphs.
• N3: Formally define a directed path Pn!, directed
cycle Cn! and directed complete graph Kn!.
Exercises 9-2
• N4: Describe a digraph that does not arise
from a binary relation..
• N5: Decribe a procedure that assings to
each simple undirected graph (V,~) a
directed graph in such a way that each
undirected path is assigned a pair of
oppositely directed arcs.
• N6: Prove that K5 can be obtained from
G(5,2) by a series of edge contractions.
10. Graph Products
Graph Products
• For two graphs G and H we may define
several graphs on the vertex set V(G) £
V(H) that behave like a product. The most
natural one is the so-called Cartesian
product that we introduce first.
Square Grid Gr(n,m).
• Let Gr(n,m) denote the graph defined by a square
grid in the plane, determined by n  m nodes. We
will call it a square grid. For instance, Gr(2,2) is
isomorphic to C4. We may regard Gr(n,m) as a
product of paths Pn and Pm. In a similar way we
may regards the n-prism graph as the product of a
cycle Cn and K2.
• This is a motivation for the introduction of the
Cartesian product of graphs.
Cartesian Product of Graphs
• The Cartesian product , G  H, of graphs G and
H has vertex set VG  VH and two vertices (u,v)
and (u’,v’) are adjacent if and only if:
• u = u’ and v ~ v’ or
• u ~ u’ and v = v’.
More products.
• The cartesian product is not the only product for
graphs. If we define a category of graphs and
dimension preserving maps, the categorical
product is the so-called tensor product. If,
however, the category admits also graph mappings
that may map edges to vertices, the categorical
product is the so-called strong product. We will
meet both of them.
Tensor Product of Graphs
• The tensor product G  H of graphs G and H has
vertex set VG  VH. Two vertices (u,v) and (u’,v’)
are adjacent if and only if:
• u ~ u’ and v ~ v’.
Strong Product of Graphs
• The Strong product G £ H of graphs G and H
has the vertex set VG  VH. Two vertices (u,v)
and (u’,v’) are adjacent if and only if:
– u ~ u’ and v ~ v’ or
– u = u’ and v ~ v’ or
– u ~ u’ and v = v’.
Examples of Products
P5 ¤ P4
P5 £ P4
P5 £ P4
• On the left, there are
cartesian, tensor and
strong products of P5
by P4. [Cartesian
product is the grid.]
Exercises 10
• N1: Let the graph Gi have ni vertices and mi edges, for i = 1,2. Determine th number
of vertices and edges in their Cartesian product G1 ¤ G2.
• N2: Let the graph Gi have ni vertices and mi edges, for i = 1,2. Determine the
number of vertices and edges in their tensor product product G1 £ G2.
• N3: Let the graph Gi have ni vertices and mi edges, for i = 1,2. Determine the
number of vertices and edges in their strong product G1 £ G2.
• N4: Under what conditions will the Cartesian product G1 ¤ G2 be connected?
• N5: Under what conditions will the tensor product G1 £ G2 be connected?
• N6: Under what conditions will the strong product G1 £ G2 be connected?
• N7: Under what conditions will the Cartesian product G1 ¤ G2 be bipartite?
• N8: Under what conditions will the tensor product G1 £ G2 be bipartite?
• N9: Under what conditions will the strong product G1 £ G2 be bipartite?
11. Factors and Factorizations
Factors
• A spanning subgraph is also called a factor. A kvalent regular factor is simply called a k-factor.
• 1-factor is a different name for perfect matching.
• A 2-factor is a disjoint union of cycles covering
the vertex set of the graph.
Factorization
• Let G = (V,E) be a graph and let E be the disjoint
union of sets E = F1 t F2 t ... t Fs, then Hi = (V,Fi)
are factors of G and the decompostion of G into
these factors is called a factorization. This is
written as:
• G = H1  H2  ...  Hs
If all factors Hi are k-factors, we speak of kfactorization of graph G.
Example.
• For any graph G and any of its spanning
subgraphs H we have the factorization:
• G = H  G\H.
Graph Power
• For a connected graph G and an integer k,
we define the k-th graph power G(k) as
follows:
• V(G(k)) := V(G).
• u ~ v if and only if d(u,v) · k.
Pure Graph Power
• For a connected graph G and an integer k,
we define the k-th pure graph power G[k] as
follows:
• V(G[k]) := V(G).
• u ~ v if and only if d(u,v) = k.
Intermezzo - Partitions, Set
Partitions, Graduated sets, etc.
•
•
•
•
•
•
•
Equivalence relation – Set Paritition
Distance Partition – Ordered Set Partition
Graduated Graph – Graduated Set
Valence Sequence – (Number) Partition
Hierarchy – Nested (Graduated) Set Partitions
Rooted Tree, Rooted Graph...
MINIVEGA should support all these structures.
Exercises 11-1
• N1. Prove: If a trivalent graph contains a 1factor, it also has a 2-factor.
• N2. Find a trivalent graph, without a 1factor.
Exercises 11-2
• N3: Prove that only regular d-valent graphs with d
divisible by k admit a k-factorization.
• N4: Prove that non-regular graphs have no 1factorizations.
• N5: Show that a 2-valent graph G has a 1-factorziation, if
and only if it is the disjoint union of even cycles.
• N6: Prove that each prism Pn has a 1-factorization.
• N7. Prove that the Petersen graph has no 1-factorization.
• N8. Show that each cubic hamiltonian graph has a 1factorization.
Exercises 11-3
• N9. Prove that for any integer k and
connected graph G on n vertices we have:
• G(k) := G[0] = En.
• G(k) = G[0] © G[1] © ... © G[k].
Homework 11
• H1. Prove that each prism graph
Pn = K2 ¤ Cn admits a 1-factorization.
• H2. Prove that each of the three products
(cartesian, tensor, strong) is associative and
commutative (up to isomoprhism).
• H3. Give a definition of the following
infinite graphs (a suitable drawing suffices):
P1, C1, T(1,k), Q1,T1, H1,
12. Planar Graphs
Planar Graphs
•
Graph G is planar, if
it can be “properly”
drawn in the plane.
In order to explain
this informal notion
we have to define
embeddings of
graphs.
Embeddings
•
•
•
Let S be a “nice” topological space such as a metric
space. An embedding of a general graph :G  S is
defined as follows:
1. Injective mapping :V(G)  S
2. Family of continuous mappings e:[0,1]  S, for
each edge e = uv so that e( 0) = (u) and e(1) =
(v).
3. In the interior of the interval e is injective and
4. its image contains no point that is an image of some
other vertex or edge.
A connected component of S – (G) is called a face of
the embedding.
Graph G is planar, if it can be embedded in the plane.
Stereographic Projection
N
T0
T1
• There is a homeomorphic
mapping of a sphere
without the north pole N
to the Euclidean plane R2.
It is called a stereographic
projection.
• Take the unit sphere
x2 + y2 + z2 = 1 and the
plane z = 0.
• The mapping
p: T0(x0,y0,z0) a T1(x1,y1)
is shown on the left.
Stereographic Projection
N
T0
T1
• The mapping
p: T0(x0,y0,z0) a
T1(x1,y1)
is shown
on the left.
• r1 = r0/(1-z0)
• x1 = x0/(1-z0)
• y1 = y0/(1-z0)
Example
• Take the
Dodecahedron and a
random point N on a
sphere.
• The associated
stereographic
projection is depicted
below.
Example
• A better strategy is to
take N to be a face
center as shown on the
left.
Euler’s formula for planar graphs
• For a connected plane graph G with v
vertices, e edges and f faces we have:
• v – e + f = 2.
• Warning: the outer face is counted!
Fary’s Theorem
• Each simple planar graph admits a specially
nice embedding.
• Theorem (Fary): Each simple planar graph
can be embedded in the plane in such a way
that all edges are represented by straight
line segments.
Kuratowski’s Theorem
• Theorem (Kuratowski): Graph G is planar
if and only if it neither contains a
subdivision of K5 nor a subdivision of K3,3.
• Graphs K5 and K3,3 are called the
Kuratowski graphs.
Applications of Kuratowski’s
Theorem
• Any graph can now be either drawn in the
plane or one can find a subdivision of a
Kuratowski graph in it.
• For any graph on n vertices there are
efficient algorithms for checking if the
graph is planar. The best one runs in linear
time ( O(n)).
Wagner’s Theorem
• Similar to Kuratowski:
• Theorem (Wagner): Graph G is planar if
and only if it contains no subgraph that can
be contracted to one of the two Kuratowski
subgraphs.
Exercises 12-1
• N1. Show that K4 can be embedded in the
plane.
• N2. Show that the Petersen graph can be
embedded in the projective plane.
• N3. Prove: A graph is planar if and only if it
can be embedded in a sphere. (Hint: use
stereographic projection.)
Exercises 12-2
• N4. By using Kuratowski’s Theorem show that the
Petersen graph is non-planar.
• N5. By using Wagner’s Theorem show that the
Petersen graph is non-planar.
• N6. Show that for any planar graph with v
vertices, e edges and girth g the following is true:
(g-2)e  g(v-2)
• N7. By using Euler’s formula show that the
Petersen graph is non-planar.
13. Graphs from Polyhedra
Skeleta of Geometric Bodies
• To each geometric polyhedron T we may
associate a graph G(T), by selecting the
vertices and edges of the polyhedron. The
obtained graph is called the skeleton of T.
• Sometimes we use the same name for the
polyhedron and for the graph. Later we will
see why such a naming is permitted.
Melancholia I
• The renowned
graphics “Melancholia
I” by Albrecht Dürer
contains a mysterious
body (polyhedron)
whose construction is
now understood.
Dürer Polyhedron
• It is obtained from an
elongated cube by
truncating the top and
bottom vertex.
• A polyhedron with 8
faces is obtained:
– 6 pentagons
– 2 regular triangles
Skeleta of Polyhedra are
Modeled by Graphs
• On the left we see the
Dürer graph, with 12
vertices and 18 edges,
the skeleton of
Dürer’s polyhedron. It
is isomorphic to the
generalized Petersen
graph G(6,2).
Steinitz Theorem
• Theorem [Steinitz]. A simple graph G is
planar and 3-connected if and only if it is
the skeleton of a convex three-dimensional
polyhedron.
Fullerene
• Fullerene is a convex
trivalent polyhedron,
whose faces are only
pentagons and
hexagons.
Dodecahedron
• The dodecahedron is
the smallest fullerene.
Buckminster Fullerene
• The most well-known
fullerene is the
Buckminster fullerene
on 60 vertices. It is a
truncated icosahedron.
• The Buckminster
fullerene is a model of
a carbon molecule.
Tutte’s Planarity Algorithm
• A cycle C of G is called peripheral if
• no edge not in C joins two vertices in C
• G \ C is connected.
• For example, a face of a 3-connected
planar graph can be shown to be a
peripheral cycle.
• The embedding r of G in the plane is
barycentric relative to S µ V(G) if for
each vertex u  S the point (vector) r(u)
is the barycenter of the of images of
neighbours of u.
Tutte’s Embedding
• Theorem [Tutte]. Let C be a peripheral
cycle of length d in a connected simple
graph G. Let s be a mapping from V(C) to
the vertices of a convex d-gon in R2 such
that adjacent vertices in C are adjacent in
the polygon. The unique barycentric
representation relative to C determines a
drawing of G in R2. This drawing has no
crossings if and only if the graph is
planar.
• [The vertex coordinates in R2 can be
obtained by solving a linear system. This
gives an O(n3) planarity test algorithm].
Prisms
• The skeleton of an n-sided prism is denoted
by Pn. It is planar, trivalent and has 2n
vertices.
Antiprisms
• The skeleton of an nsided antiprism is
denoted by An. Its
graph is planar,
tetravalent and has 2n
vertices.
A6
Möbius Ladders
• Möbius ladder Mn is
obtanied from C2n by
adding n main
diagonals.
M5
Exercises 13-1
• N1. The angles in the Dürer
pentagon (see figure on the
left) indicate fivefold symmetry
and thus implicitely the golden
section. Determine the lengths
of the sides of Dürer
polyhedron.
• N2. Determine coordinates of
the Dürer polyhedron. (Hint: the
pentagon on the left has a
circumscribed circle.)
Exercises 13-2
• N3. Prove that each fullerne has
exactly 12 pentagonal faces.
• N4. Prove that each fullerene
has an even number of vertices.
• N5. Prove that a fullerene has
at least 20 vertices.
• N6. Show that for each even n,
n ¸ 20, n  22 there exists a
fullerne on n vertices.
Exercises 13-3
• N7. Show that Pn can be obtianed from Mn
by deleting and re-attaching two edges.
• N8. Use Tutte’s algorithm to draw the
Petersen graph and use one of the
pentagonal cycles C5 as the peripheral
regular pentagon with unit side length.
Verify that the inner cycle is drawn as the
pentagram. Determine its side-length.
Homework 13
• H1. Show that every graph can be embedded in R3
(Hint: place vertices on the helix curve (cos t, sin
t, t) or even better on the curve (t,t2,t3).
• H2. Show that there is only one fullerene on 24
vertices. Draw its skeleton.
• H3. Prove that none of the Möbius ladders Mn is
planar.
• H4. Use Tutte’s algorithm to draw the cube Q3 and
use the unit square for the outer face. What is the
length of a side in the opposite, inner face?
14. Metric Space - Revisited
Metric Space - Revisited
• If (M,d) is a metric space, then for any
A
µ M with induced metric (A,d) is also a
metric space, namely a subspace.
• A natural question is : when are two metric
spaces (M,d) and (M’,d’) considered
isomorphic?
• There are two types of mappings that are
candiates for “isomorphism”.
Isometries
• Let (M,d) and (M’,d’) be two metric spaces.
A bijective mapping s: M ! M’ is called
isometry, if for every pair of points u,v 2 M
we have:
• d(u,v) = d’(s(u),s(v)).
• Clearly, isometric spaces are
indistingushable as far as metric properties
are concerned.
Euclidean metric in Rn.
• The set of real n-tuples
Rn := {x = (x1,x2,...,xn)|xi 2 R, 1
· i · n}
• carries a number of important mathematical
structures. The mapping
• dp(x, y) = [(x1 – y1)p + (x2 – y2)p + ... + (xn – yn)p]1/p.
• makes (Rn,dp) a metric space for 1 · p · 1.
• For p = 2 the usual Euclidean metric is
obtained.
Metric in C.
• Let z = a + bi and w = c + di be two
complex numbers.
• Define d(z,w) := |z –w|. Then (C,d) is a
metric space.
• Note that (C,d) is isometric to the Euclidean
plane (R,d2).
Similarity I
• Let (M,d) and (M’,d’) be two metric spaces.
A mapping h:M ! M’ with the property that
for any four points a,b,c,d 2 M we have:
• If d(a,b) = d(c,d) then d(h(s),h(b)) =
d(h(c),h(d)) is called similarity (of type I).
Similarity II
• Let (M,d) and (M’,d’) be two metric spaces
and r 2 R\{0}. A mapping h:M ! M’ with
the property that for any pair of points a,b, 2
M we have:
• If d(a,b) = r d(h(a),h(b)) then h is called
similarity (of type II) and r is called the
dilation factor.
Type I vs. Type II
• Clearly each similarity of type II is also a
similarity of type I. In general, the converse
is false.
• Theorem. A similarity on (Rn,d2) of type I
is also of type II. (Proof can be found in
Paul B. Yale: Geomerty and Symmetry,
Dover, 1988 (reprint from 1968))
Finite Metric Space
• In a finite metric space (M,d) we may
assume that min d(u,v) = 1. Max d(u,v) is
called the diameter of M. The quotient Max
d(u,v)/Min d(u,v) is called dilation
coefficient.
15. Representations of Graphs
Representation of Graphs
• Let G be a graph and let V be a set. A pair of mappings
rV:V(G) ! V and rE:V(G) ! P(V) is called a Vrepresentation of graph G if for any edge e = uv 2
E(G) we have {rV(u),rV(v)} µ rE(uv). If there is no
danger of confusion we will drop the subscripts and
denote both mappings simply by r.
• Usually we require V to be a vector space (this is what
C. Godsil and G. Royle do in their book Algebraic
Graph Theory, Springer, 2001). But that is not always
the case. In their definition Godsil and Royle use a
single mapping defined on the vertices. In such a case
we may extend the mapping on the edge set in an
arbitrary way, for instance by taking rE(uv) :=
{rV(u),rV(v)}.
Representation of Graphs in a
Metric Space
• There are important and deep results by László
Lovász et al.
• Sometimes we may take V to be a metric space,
projective space or some other structure.
• If (V,d) is a metric space we may define the
energy of the representation.
Point Configuration
• A point configuration S µ V
is a collection of elements of
some space V. Later we will
consider point configurations
in R2.
• If r is a V-representation of G
then the image S = r(V(G)) is
a point configuration.
• We say that r is vertex
faithful is r:V(G) ! S is a
bijection. We are mostly
interested in vertex faithful
representations.
Graph Representation – An
Example
• For the cube graph Q3 there are
several useful representations:
• [3 dimensional real representation] In
R3 the eight vertices are mapped to
the eight points of {0,1}3.
• The two drawings of Q3 in the
Euclidean plane can be interpreted
as representations in
• [2 dimensional real representation]
R2 or in
• [1 dimensional complex
representation] C.
• In the latter case, the points in the
complex plane are given by
{eikp|0  k  7}.
Extending Representation to
Edges
• Usually we try to extend the mapping r to the edges. In
the case V = R2 or V= R3 finding a representation means
actually drawing graph G in V = R2 or V= R3 .
• Each edge e=uv is then represented as the line segment
connecting r(u) and r(v).
• Hence r(e) = conv(r(u),r(v)).
• In general we extend r to the edges r: E(G) ! P(V)
and require that for e = uv, {r(u),r(v)} µ r(e).
• If nothing is said about edge extension, we assume r(e)
= {r(u),r(v)}.
Edge Extensions
r(u)
r(u)
r(u)
r(u)
r(u)
r
r
r(v)
r(v)
• Let e = uv 2 E(G).
• There are several possible edge
extensions:
• r(e) = {r(u),r(v)}.
• r(e) = {r(u),r,r(v)}.
• r = (r(u)+r(v))/2.
r(v)
r(v)
r(v)
• r(e) = conv(r(u),r(v)).
• r(e) = aff(r(u),r(v))
• We may speak of barycentric,
convex and affine edge
extensions, respectively.
• But there are several other
interpretations of r.
Three Classical Results
• The Steinitz Theorem, Fary’s Theorem and
Tutte’s Theorem can all be interpreted as
graph representations.
Graph Representation vs. Graph
Drawing
• There is some overlap but there are many differences.
• In graph drawing (in the broad sense of the word) the object is to
find algorithms to draw a graph (usually in the plane) with
certain restrictions or with some optimization criterion.
[Computer Science Approach.] See for example: Annotated
bibliography on graph drawing algorithms, by Di Battista,
Eades, Tamassia and Tollis.
• In graph representation we label vertices (= add coordinates). We
may look at this as a functor from the category of graphs to the
category of coordinatized graphs. [Mathematical Approach].
• We will use the word graph drawing in a narrow sense of the
word.
The Energy
• Usually we try to find among the
representations of certain type the one that
is “optimal” in a cetrain sense.
• To this end we may define an energy
function E(r) and then seek a representation
that minimizes the energy.
• There are several such energy functions
used in various problem areas.
Some Energy Models
•
•
•
•
Spring embedders
Molecular mechanics
Tutte drawing
Schlegel diagram drawing
(B. Plestenjak).
• [Connection to Markov
Chains]
• ...
• Laplace Representation
The Laplace Representation
•
Let r be a representation in Rk.
Define
E(r) = Suv 2 E(G) ||r(u)-r(v)||2
An R3 Laplace representation of a
fullerene (skeleton of a trivalent
polyhedron with pentagonal and
hexagonal faces)
•
It turns out that the minimum (under
some reasonable conditions) is
achieved as follows.
1.
2.
3.
4.
Take the Laplace matrix of G.
Q(G) = D(G)-A(G)
Find the eigenvalues
0 = l1 · l2 · ... · ln.
Find the corresponding orthonormal eigenvectors
x1, x2, ..., xn.
Form a matrix R =[x2|x3| ... |xk+1]
Let r(vi) = rowi(R).
5.
6.
Nodal Domains
• A one dimensional
representation defines a
partition of the vertex set
into three classes: V+, V-,
V0.
• A nodal domain is a
connected component of
the graph induced by V+or
V-. [Weak nodal domain
V+ [ V0].
Nodal Domains
• The Example on the
left represents nodal
domains obtained
from the Laplace
representation of
G(10,4).
Congruence and Similarity
• A representation in any metric sapce, in
particular in Rn, can be scaled without
“being changed too much”. If r is injective
on the vertices, we may scale it in such a
way that Min d(u,v) = 1, for u ~ v. Each
vertex faithful representation is similar to a
standard one.
Unit Distance Graphs
• Let r be a representation in Rk. Define
Ep (r) = (Suv 2 E(G) ||r(u)-r(v)||p) (1/p)
• We assume that Min uv 2 E(G) ||r(u)-r(v)|| = 1
• In the limit when p ! 1 we get
E1 (r) = Maxuv 2 E(G) ||r(u)-r(v)||
• The number E1 (r) is called dilation coefficient.
• Hence E1 (r) ¸ 1. In the special case: E1 (r) = 1 we
call this representation a unit distance graph.
Flat Torus
r = (rx,ry)
s = (sx,sy)
• Take a unit square and identify
two opposite pairs of sides. The
resulting topological space is a
torus. In order to make it a metric
sapce we can extend the usual
Euclidean distance .
• dT(r,s) := Min{d(r,s+(0,1)),
d(r,s+(1,0)), d(r,s+(1,1)),
d(r,s+(0,-1)). d(r,s+(-1,0)),
d(r,s+(-1,1)), d(r,s+(1,-1)),
d(r,s+(-1,-1))}.
Embeddings vs. Representations
•
Let S be a “nice” topological space such as a metric
space and G be a general graph. Let a mapping :G  S
be defined having the following properties:
1. :V(G)  S is injective.
2. For each edge e=uv e:[0,1]  S, is continuous and
e( 0) = (u) and e(1) = (v).
3. In the interior of the interval [0,1] e is injective.
•
•
Each embedding would qualify.
Note that  defines a representation of G in S.
Embeddings are Representations
• Think of K3 ¤ K3
embedded in the torus.
The torus, in turn, is
embedded in R3. We
obtain a representation
of our graph in the
torus and another one
in R3.
Stereographic Projection
N
T0
T1
• There is a homeomorphic
mapping of a sphere
without the north pole N
to the Euclidean plane R2.
It is called a stereographic
projection.
• Take the unit sphere
x2 + y2 + z2 = 1 and the
plane z = 0.
• The mapping
p: T0(x0,y0,z0) a T1(x1,y1)
is shown on the left.
Stereographic Projection
N
T0
T1
• The mapping
p: T0(x0,y0,z0) a
T1(x1,y1)
is shown
on the left.
• r1 = r0/(1-z0)
• x1 = x0/(1-z0)
• y1 = y0/(1-z0)
Stereographic projection and
representations
• We may use stereographic projection to get
an R2 drawing from an R3 drawing.
• Note that the representation of edges is
computed anew!
Example
• Take the Dodecahedron
and a random point N on a
sphere.
• The corresponding
stereographic projection is
depicted below.
• A better strategy is to take
N to be a face center.
Example
• A better strategy is to
take N to be a face
center as shown on the
left.
• Only vertices are
projected. The edges
are re-computed.
Schlegel Diagram
• Schlegel diagram are
defined for (convex)
polyhedra. Normally, a
Schlegel diagram is
definded as a projection of
the polyhedron on one of
its faces.
• We understand this notion
in a broader sense, namely
as a drawing of a graph G
within the convex region
defined by some of its
vertices S ½ V(G).
Exercises 15-1
• N1. Given a standard drawing of G(n,r)
with inner radius r and outer radius R,
determine the dilation coefficient of this
planar representation.
• N2. Select the optimal quotient R/r in the
previous exercise.
• N3. In the unit flat torus draw the circle of
radius ½ centered at the point (¼, ¼).
Exercises 15-2
• N4. Use Laplace representations followed by
stereographic projection to get Schlegel diagrams
of platonic graphs.
• N5. Use a generalization of Tutte’s method to
solve the same problem.
• N6. Repeat the two exercises for some of the
archimedean solids and their duals.
• N7. Is there a unit distance representation for the
subdivision graph S(K4)?
Homework 15
• H1. It is easy to verify that K4 is not a unit
distance graph in the plane. Consider a
drawing of K4 in the plane with only two
distinct edge lengths. How many such nonisomorphic drawings are there? (Hint: there
are six). Compute the dilation coefficient
for all such drawings.
16. Edge-Coloring and Snarks
Edge-Coloring of Graphs
• On the left we see a 1factorization of P5, the five-sided
prism. Each factor is respresented
by its own color.
• No edges of the same color are
incident with the same vertex. In
other words, each set of
monochromatic edges is
independent.
• This idea can be formalized.
Edge-Coloring of Graphs
• A mapping c:E(G)  C from
the edge set E(G) to some finite
set C is called an (admissible)
edge coloring, if for any two
edges e and f with a common
endvertex c(e)  c(f).
• The least number of colors
needed to properly color the
edges of G is called chromatic
index and is denoted by c’(G).
Vizing’s Theorem
• Theorem: The chromatic index of a simple graph
G satisfies the following inequalities:
• D(G)  c’(G)  D(G)+1
• Proof. Lower bound immediate, the upper bound
is more difficult to prove.
• Using Vizing’s theorem we may now classify
simple graphs into two types. A graph of type I has
c’(G) = D(G), while the graph of type II has c’(G)
= D(G)+1.
Consequences
• Corollary: A regular graph is of type I, if
and only if it has a 1-factorization.
• Example: The Petersen graph is of type II.
• Even if we replace a vertex by a triangle in
the Petersen graph, the resulting graph
remains of type II.
Triangle Removal
a
c
• Let G and G’ be two trivalent graphs
that differ only by a triangle.
• This means that G’ is obtained from
G by replacing any vertex by a
triangle as shown on the left.
• Equivalently, G is obtained from G’
by a triangle removal.
• Proposition. If G and G’ differ by a
triangle then c’(G) = c’(G’).
b
Vertex is replaced by triangle.
a
c
b
Snarks
• Using a similar
argument we may
prove that cycles of
length 4 can be
removed in the same
sense.
• A 3-connected
trivalent graph of girth
> 4 of type II is called
a snark.
Families of Snarks
• There are several
infinite families of
snarks known.
• Blanuša found the first
snark after Petersen.
Blanuša snarks
König’s Theorem
• Theorem (König): For a bipartite graph G
we have c’(G) = D(G).
Exercises 16
• N1. Show that there are no bipartite snarks.
• N2. Repeatedly remove all quadrilaterals
and triangles in K3,3. What simple graph is
obtained in the end?
• N3. Dot product!
17. Vertex Colorings and Graph
Homomorphisms
Vertex Coloring of Graphs.
• A mapping c:V(G)  C from the vertex set to a
finite set of colors is called vertex coloring, if for
any pair of adjacent vertices u ~ v we have c(u) 
c(v).
• The least number of colors of some proper vertex
coloring of G is called the chromatic number
and is denoted by c(G).
Brooks’ Theorem
• For any connected graph G the following
holds:
 c(G) = D(G)+ 1, if G is isomorphic to a complete
graph or an odd cycle.
• c(G) · D(G), otherwise.
Four Color Theorem
• A theorem posed in the 19th century and
proved in the 20th century.
• Theorem: c(G) · 4, for any planar graph G.
Graph Mappings
(Homomorphisms)
• Let f:V(G) ! V(H) be a mapping between
the vertices of two graphs. f is called a
graph mapping or graph homomorphism
if for any pair of vertices u,v 2 V(G) the
fact u ~ v implies f(u) ~ f(v).
More general maps – Weak
homomorphism
• Sometimes we allow graph maps that do not
preserve dimension. f: V(G) ! V(H) and u
~ v implies f(u) ~ f(v) or f(u) = f(v).
• Such a f is called a weak homomorphism.
Retracts
• Let f: G ! G be a graph homorphism. Then
f is called a retraction if f2 = f.
(idempotent). f(G) = H has the property that
f|H = id. H is called a retract.
• In a similar way one can define a weak
retraction and weak retract.
Colorings revisited.
• A vertex coloring c with h colors can be
defined as a graph homomorphism
c:G ! Kh
Exercises 17
• N1. Prove that retracts and weak retracts are
isometric subgraphs.
• N2(*). A graph G is called a median graph
if for any triple of vertices u,v, w we have
|I(u,v) Å I(v,w) Å I(w,u)| = 1. Prove that G is
a median graph if and only if it is a retract
of some hypercube.
Statistic page
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Number of slides:226
Number of sections:17
Number of exercises:112
Number of homeworks:18