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Super edge-graceful labelings for total stars and total cycles

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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

- 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

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).

Edge-graceful labeling

p=4 So vertex labels are 0, 1, 2, 3

q=5 So edge labels are 1, 2, 3, 4, 5

4

1 3 5

2

An Edge-graceful labeling for K4 minus an edge

1 4 0

1 3 5

2 2 3

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).

6

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.

Corollary: A complete graphs on p vertices is not edge-graceful, if p≡ 2 (mod 4).

Corollary: A complete bipartite graph Km,m

is not edge-graceful.

Corollary: Petersen graph is not 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.

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.

A New Labeling

4

-2

2

-1

1

-3

-4

3

A New Labeling

2

4

-2

1

2

-1

-3

3

1

-3

-2

-4

3

-1

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/2if q is even and with

0, ±1, ±2,…,±(q-1)/2if q is odd.

If the vertex sums are

±1, ±2,…,±p/2when p is even and

0, ±1, ±2,…,±(p-1)/2when p is odd,

then G is super edge-graceful.

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 Pnexcept P2and P4and all cycles

except C4and C6are super edge-graceful.

A. Khodkar, R. Rasi and S.M. Sheikholeslami (2008):

The complete graph Kn issuper edge-gracefulfor

all n ≥ 3, n ≠ 4.

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.

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.

Stars

Star with 5 vertices: St(5)

18

Total Stars

T(St(5))

19

Total Stars

-2

T(St(5))

5

6

1

-4

3

-3

4

-1

-5

-6

2

Edge Labels: ±1, ±2, ± 3, ± 4, ± 5, ± 6

Vertex Labels: 0, ±1, ±2, ± 3, ± 4

20

SEGL for T(St(2n+1))

SEGL for T(St(9))

Edge Labels: ±1, ±2, ± 3, … , ± 12

Vertex Labels: 0, ±1, ±2, ± 3, …, ± 8

21

SEGL for T(St(2n))

SEGL for T(St(10))

Edge Labels: 0, ±1, ±2, ± 3, … , ± 13

Vertex Labels: 0, ±1, ±2, ± 3, …, ± 9

Total cycle T(C8)

Edge Labels: ±1, ±2, ± 3, … , ± 12

Vertex Labels: ±1, ±2, ± 3, …, ± 8

4

-4

-12

12

5

3

-5

-8

8

-3

11

-11

-6

6

-1

2

-2

1

10

-10

-7

7

9

-9

SEGL of total cycle T(C8)

SEGL of total cycle T(Cn)

SEGL for T(St(16))

Edge Labels: ±1, ±2, ± 3, … , ± 24

Vertex Labels: ±1, ±2, ± 3, …, ± 16

SEGL for T(St(16))

SEGL of total cycle T(Cn)), n ≡0 (mod 8)

SEGL for the Union of Vertex Disjoint of 3-Cycles

0

3

3

-2

2

-1

4

1

-4

-4

4

-2

-1

1

2

3

-3

0

Edge labels and vertex labels are 0, ±1, ±2, ±3, ±4

SEGL for the Union of Vertex Disjoint of 3-Cycles

-6

5

-4

6

-3

-4

1

3

-2

-1

5

2

-1

-2

1

2

-5

-3

3

4

4

-6

-5

6

Edge labels and vertex labels are ±1, ±2, ±3, ±4, ±5, ±6

Let a + b + c = 0.

-b

a

c

-c

-a

b

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.

An Open Problem: Super edge-gracefulness of disjoint union of four cycles.

Edge Labels=Vertex Labels={1, -1, 2, -2}

0

-1

-1

1

2

1

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}

-1

1

-3

-4

3

4

2

3

-2

-3

-1

-2

1

2

-4

4

Hence, the disjoint union of two 4-cycles is SEG.

34

Is the disjoint union of three 4-cycles SEG?

Edge Labels=Vertex Labels={±1, ±2, ±3, ±4, ±5, ±6}

35

An Open Problem: The disjoint union of m 4-cycles is super edge-graceful if m>3.

36

37

Thank You

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