Transcript On k-Edge-Magic Halin Graphs

On k-Edge-magic Cubic Graphs
Sin-Min Lee, San Jose State University
Hsin-hao Su*, Stonehill College
Yung-Chin Wang, Tzu-Hui Institute of Technology
24th MCCCC
At
Illinois State University
September 11, 2010
Supermagic Graphs

For a (p,q)-graph, in 1966, Stewart defined
that a graph labeling is supermagic iff the
edges are labeled 1, 2, 3, …, q so that the
vertex sums are a constant.
k-Edge-Magic Graphs


A (p,q)-graph G is called k-edge-magic
(in short k-EM) if there is an edge
labeling l: E(G)  {k, k+1, …, k+q-1}
such that for each vertex v, the sum of the
labels of the edges incident with v are all
equal to the same constant modulo p; i.e.,
l+(v) = c for some fixed c in Zp.
If k =1, then G is said to be edge-magic.
Examples: 1-Edge-Magic

The following maximal outerplanar
graphs with 6 vertices are 1-EM.
Examples: 1-Edge-Magic

In general, G may admits more than
one labeling to become a k-edge-magic
graph with different vertex sums.
Examples: k-Edge-Magic

In general, G may admits more than one
labeling to become a k-edge-magic graph.
Necessary Condition

A necessary condition for a (p,q)-graph G
to be k-edge-magic is
qq  2k  1  0mod p 

Proof:


qk  k  q  1
2
The sum of all edges is
Every edge is counted twice in the vertex
sums.
k-Edge-Magic is periodic

Theorem: If a (p,q)-graph G is k-edgemagic then it is pt+k-edge-magic for
all t ≥ 0 .
Cubic Graphs

Definition: 3-regular (p,q)-graph is
called a cubic graph.
The relationship between p and q is
3p
q
2

Since q is an integer, p must be even.

One for All


Theorem: If a cubic graph is k-edgemagic, then it is k-edge-magic for all k.
Proof:

Since every vertex is of degree 3, by adding or
subtracting 1 to each adjacent edge, the vertex
sum remains the same.
Examples: Complete Bipartite

The complete bipartite graph K3,3 is kedge-magic for all k.
Not Order 4s


Theorem: A cubic graph with order 4s is
not k-edge-magic for all k.
Proof:


The number of edges is 6s.
The necessary condition implies
0  6s6s  2k  1  2smod4s .

It is impossible for all k.
Möbius Ladders


The concept of Möbius ladder was
introduced by Guy and Harry in 1967.
It is a cubic circulant graph with an even
number n of vertices, formed from an ncycle by adding edges (called “rungs”)
connecting opposite pairs of vertices in the
cycle.
Möbius Ladders

A möbius ladder ML(2n)
with the vertices denoted
by a1, a2, …, a2n. The
edges are then {a1, a2},
{a2, a3}, … {a2n, a1}, {a1,
an+1}, {a2, an+2}, … , {an,
a2n}.
Labeling Idea


Splits all edges into two subsets. The
first subset contains all the edges of C2n.
The second set contains all middle
edges, which forms a perfect matching.
Construct a graceful labeling for the first
subset, i.e., an arithmetic progression.
The rest numbers also form an
arithmetic progression.
Labeling Method 1


Divides the numbers into three subsets:
{0, 1, 2, 3, …, 2k = n-1}, {2n, 2n+1,
2n+2, 2n+3, …, 2n+2k = 3n-1}, {n, n+1,
n+2, n+3, …, n+2k = 2n-1}.
Use the first two subsets to label C2n by
the following sequence: k+1, 1, k+2, 2,
k+3, 3, …, 2k, k, 0, k+1, 1, k+2, 2, k+3,
3, …, 2k, k, 0.
Example of Method 1
Labeling Method 2


We label the edges by 1, 1, 2, 2, 3, 3, …,
k+1, k+1, n+k+2, k+2, n+k+3, k+3, …,
2n, 2k+1.
Label the rest numbers, k+2, k+3, …,
n+k+1 to the edges in the middle.
Example of Method 2
Cylinder Graphs

Theorem (Lee, Pigg, Cox; 1994): The
cylinder graph CnxP2 is a 1-edge-magic graph
if n is odd.
Cylinder Graphs Examples
Generalized Petersen Graphs



The generalized Petersen graphs P(n,k) were
first studied by Bannai and Coxeter.
P(n,k) is the graph with vertices {vi, ui : 0 ≤ i
≤ n-1} and edges {vivi+1, viui, uiui+k}, where
subscripts modulo n and k.
Theorem: The generalized Petersen graph
P(n,t) is a k-edge-magic graph for all k if n is
odd.
Gen. Petersen Graph Ex.
Order 6

Theorem: A cubic graph with order 6 is kedge-magic for all k.
Order 10
Order 14: Transformation
Order 14: Transformation
Conjecture


Conjecture: A cubic graph with order
4s+2 is k-edge-magic for all k.
With the previous examples, this is a
reasonable extension of a conjecture by
Lee, Pigg, Cox in 1994.
Mod(m)-Edge-Magic Graphs

A (p,q)-graph G is called Mod(m)-edgemagic (in short Mod(m)-EM) if there is
an edge labeling l: E(G)  {1,2,…,q}
such that for each vertex v, the sum of the
labels of the edges incident with v are all
equal to the same constant modulo m;
i.e., l+(v) = c for some fixed c in Zm.
Relationship between EM


Theorem: For a graph with order p, if it is
1-edge-magic, then it is mod(m)-edgemagic for m to be a factor of p.
Proof:

Since m is a factor of p, the constant sum in Zp
remains constant in Zm.
Counterexample
Proof

Since it is mod(5)-edge-magic, we have
relations as followings:




a + b = l + m,
b + c = k + l,
h + i = f + e,
(1)
(2)
(3)
From relations (1) and (2), we have

a + k = c + m.
(4)
Proof (continued)

Therefore we have g = d. Then we have a
new relation


(5)
From relations (3) and (5), we have


h + f = i + e.
i = f.
Then we have


h = e, and
g = j = d.
Proof (continued)



Without losing generality, we say d = 0,
e = 1 and f = 4.
From relation (4), we have a = 1, k = 4,
c = 2, m = 3 or a = 1, k = 4, c = 3, m = 2
or a = 2, k = 3, c = 1, m = 4 or a = 2, k =
3, c = 4, m = 1.
Here, we already run out of 1 and 4 and
only 2 and 3 left in the set.
Proof (continued)



For the case a = 1, we have b = 2 and n =
2. If forces that o = 3 and l = 3. But m = 3
or 2 can’t make the sum on v9 equal to 0.
This is a contradiction.
With the same argument, we can show
that all the possibilities can’t be true.
Therefore it is not mod(5)-edge-magic.
Future Problems



Do we have just a few counterexamples?
Any better necessary condition?
Possible sufficient conditions?