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

5

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.

A New Labeling

4

1

2

-4

-3

3

-2

-1

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))

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-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

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-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