Transcript Super edge graceful labelings for total stars and total cycles

Super edge-graceful labelings for
total stars and total cycles
Abdollah Khodkar
Department of Mathematics
University of West Georgia
www.westga.edu/~akhodkar
Joint work with: Kurt Vinhage, Florida State University
Overview
1. Edge-graceful labeling
2. Super edge-graceful labeling
3. Super edge-graceful labeling
of total stars
4. Super edge-graceful labeling
of total cycles
5. An open problem
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Edge-graceful labeling
S.P. Lo (1985) introduced edge-graceful labeling.
A graph G of order p and size q is edge-graceful if the
edges can be labeled by 1, 2, … , q such that the vertex
sums are distinct (mod p).
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Edge-graceful labeling
p=4
q=5
So vertex labels are 0, 1, 2, 3
So edge labels are 1, 2, 3, 4, 5
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1
3
5
2
4
An Edge-graceful labeling for K4 minus an edge
1
1
2
4
3
0
5
2
3
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Theorem: (Lo 1985)
A necessary condition for a graph of order p and size q to
be edge-graceful is that p divides
(q2+q-(p(p-1)/2)).
That is, q(q +1) ≡ p(p-1)/2 (mod p).
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Corollary: No cycle of even order is edge-graceful.
Proof: In a cycle of order p we have q=p. By the
Theorem, p divides q2+q-(p(p-1)/2)=p2+p-(p(p-1)/2).
Therefore,
p(p-1)/2=kp for some positive integer k. This implies
p=2k+1.
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Corollary: There is no edge-graceful tree of even order.
Proof: Let p=2k, then q=2k-1.
So (2k-1)(2k)-2k(2k-1)/2=2km.
Hence, 2k-1=2m, a contradiction.
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Corollary: A complete graphs on p vertices is not edgegraceful, if p ≡ 2 (mod 4).
Corollary: A complete bipartite graph Km,m
is not edge-graceful.
Corollary: Petersen graph is not edge-graceful.
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Theorem: Lee, Lee and Murty (1988)
If G is a graph of order p ≡ 2 (mod 4), then G is not
edge-graceful.
Conjecture: Kuan, Lee, Mitchem and Wang (1988)
Every odd order unicyclic graph is edge-graceful.
Conjecture: Sin-Min Lee (1989)
Every tree of odd order is edge-graceful.
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A New Labeling
4
-2
2
-1
1
-3
-4
3
A New Labeling
2
4
1
2
3
1
-2
-1
-3
-2
-4
3
-1
-3
Super edge-graceful labeling
J. Mitchem and A. Simoson (1994):
Consider a graph G with p vertices and q edges.
We label the edges with
±1, ±2,…,±q/2 if q is even and with
0, ±1, ±2,…,±(q-1)/2 if q is odd.
If the vertex sums are
±1, ±2,…,±p/2 when p is even and
0, ±1, ±2,…,±(p-1)/2 when p is odd,
then G is super edge-graceful.
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J. Mitchem and A. Simoson (1994): If G is
super edge-graceful and
p | q, if q is odd, or
p | q+1, if q is even,
then G is edge-graceful.
Theorem: Super edge-graceful trees of odd order
are edge-graceful.
S.-M. Lee and Y.-S. Ho (2007): All trees of odd
order with three even vertices are
super edge-graceful.
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S. Cichacz, D. Froncek, W. Xu and A. Khodkar (2008):
All paths Pn except P2 and P4 and all cycles
except C4 and C6 are super edge-graceful.
A. Khodkar, R. Rasi and S.M. Sheikholeslami (2008):
The complete graph Kn is super edge-graceful for
all n ≥ 3, n ≠ 4.
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A. Khodkar, S. Nolen and J. Perconti (2009):
All complete bipartite graphs Km,n are super edge-graceful
except for K2,2, K2,3, and K1,n if n is odd.
A. Khodkar (2009):
All complete tripartite graphs are super edge-graceful
except for K1,1,2.
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A. Khodkar and Kurt Vinhage (2011):
Total stars and total cycles are super edge-graceful.
Lee, Seah and Tong (2011):
Total cycles (T(Cn)) are edge-graceful if and only if n is
even.
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Stars
Star with 5 vertices: St(5)
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Total Stars
T(St(5))
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Total Stars
-2
T(St(5))
5
-4 6
1
3
-3
-6 4
-1 -5
2
Edge Labels: ±1, ±2, ± 3, ± 4, ± 5, ± 6
Vertex Labels: 0, ±1, ±2, ± 3, ± 4
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SEGL for T(St(2n+1))
SEGL for T(St(9))
Edge Labels: ±1, ±2, ± 3, … , ± 12
Vertex Labels: 0, ±1, ±2, ± 3, …, ± 8
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SEGL for T(St(2n))
SEGL for T(St(10))
Edge Labels: 0, ±1, ±2, ± 3, … , ± 13
Vertex Labels: 0, ±1, ±2, ± 3, …, ± 9
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Total cycle T(C8)
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Edge Labels: ±1, ±2, ± 3, … , ± 12
Vertex Labels: ±1, ±2, ± 3, …, ± 8
-4
4
-12
12
3
-5
11
-6
5
-8
-3
8
-1
2
-11
-2
1
10
-7
6
-10
7
9
-9
SEGL of total cycle T(C8)
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SEGL of total cycle T(Cn)
SEGL for T(St(16))
Edge Labels: ±1, ±2, ± 3, … , ± 24
Vertex Labels: ±1, ±2, ± 3, …, ± 16
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SEGL for T(St(16))
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SEGL of total cycle T(Cn)), n ≡ 0 (mod 8)
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SEGL for the Union of Vertex Disjoint of 3-Cycles
0
3
-1
-4
-3
-2
2
4
1
2
3
1
-2
0
4
-4
3
-1
Edge labels and vertex labels are 0, ±1, ±2, ±3, ±4
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SEGL for the Union of Vertex Disjoint of 3-Cycles
5
6
1
-5
5
-6
3
2
-1
-3
-5
-6
-4
-3
-1
-2 3
4
1
-4
-2
4
6
2
Edge labels and vertex labels are ±1, ±2, ±3, ±4, ±5, ±6
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Let a + b + c = 0.
-b
a
c
-c
-a
b
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A. Khodkar (2013):
The union of vertex disjoint 3-cycles is super edge-graceful.
Example: The union of fifteen vertex disjoint 3-cycles is
Super edge graceful.
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An Open Problem: Super edge-gracefulness of disjoint
union of four cycles.
Edge Labels=Vertex Labels={1, -1, 2, -2}
0
-1
1
1
-1
2
3
1
Hence, C4 is not super edge-graceful.
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Disjoint union of two 4-cycles
Edge Labels=Vertex Labels={1, -1, 2, -2, 3, -3, 4, -4}
-3
-1
-2
2
1
-4
3
-3
2
3
1
-2
-1
4
4
-4
Hence, the disjoint union of two 4-cycles is SEG.
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Is the disjoint union of three 4-cycles SEG?
Edge Labels=Vertex Labels={±1, ±2, ±3, ±4, ±5, ±6}
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An Open Problem: The disjoint union of m 4-cycles is
super edge-graceful if m>3.
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Thank You
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