abdollah khodkar department of mathematics university of west georgia akhodkar joint work with: kurt...
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Abdollah KhodkarDepartment of Mathematics
University of West Georgia
www.westga.edu/~akhodkar Joint work with: Kurt Vinhage, Florida State University
Super edge-graceful labelings for total stars and total cycles
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Overview
1. Edge-graceful labeling
3. Super edge-graceful labeling of total stars
2. Super edge-graceful labeling
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 So vertex labels are 0, 1, 2, 3q=5 So edge labels are 1, 2, 3, 4, 5
4
1 3 5
2
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An Edge-graceful labeling for K4 minus an edge
1 4 0
1 3 5
2 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 impliesp=2k+1.
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Proof: Let p=2k, then q=2k-1.
So (2k-1)(2k)-2k(2k-1)/2=2km.
Hence, 2k-1=2m, a contradiction.
Corollary: There is no edge-graceful tree of even order.
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Corollary: A complete graphs on p vertices is not edge-graceful, if p ≡ 2 (mod 4).
Corollary: Petersen graph is not edge-graceful.
Corollary: A complete bipartite graph Km,m
is not edge-graceful.
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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.
Theorem: Lee, Lee and Murty (1988)If G is a graph of order p ≡ 2 (mod 4), then G is not edge-graceful.
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A New Labeling
4
1
2
-4
-3
3
-2
-1
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A New Labeling
4
1
2
-4
-3
3
-2
-1
2
-1
-23 -3
1
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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 with0, ±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.
S.-M. Lee and Y.-S. Ho (2007): All trees of odd order with three even vertices are super edge-graceful.
Theorem: Super edge-graceful trees of odd order are 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 forall 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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StarsStar with 5 vertices: St(5)
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Total Stars
T(St(5))
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Total Stars
T(St(5))
56
-3
-2
1
4
Edge Labels: ±1, ±2, ± 3, ± 4, ± 5, ± 6
Vertex Labels: 0, ±1, ±2, ± 3, ± 4
-5 -6
2
3-4
-1
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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(10))
SEGL for T(St(2n))
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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SEGL of total cycle T(C8)
Edge Labels: ±1, ±2, ± 3, … , ± 12
Vertex Labels: ±1, ±2, ± 3, …, ± 8
-8
2
8
1 -2
-3
-1
4
3
-12
5
6
-11
7
-10
9 -9
10
-412
11
-5
-6
-7
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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
-3 30
2 -2 1 -4-1 4
03
-2 4
3
-4 1 2 -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
Edge labels and vertex labels are ±1, ±2, ±3, ±4, ±5, ±6
6
-5 -1
5 -6-4
13-2-3 24
1
-6
-45 2 -1 -2-3
-5
3
64
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c
-b
-a -c
a
b
Let a + b + c = 0.
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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 isSuper edge graceful.
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An Open Problem: Super edge-gracefulness of disjoint union of four cycles.
-1
1
1
-1
2
01
3
Edge Labels=Vertex Labels={1, -1, 2, -2}
Hence, C4 is not super edge-graceful.
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-1
-44
1
2
-3 3
-2
Edge Labels=Vertex Labels={1, -1, 2, -2, 3, -3, 4, -4}
2
-2
1
3
-1
4
-3
-4
Hence, the disjoint union of two 4-cycles is SEG.
Disjoint union of two 4-cycles
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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