on k-edge-magic halin graphs sin-min lee, san jose state university hsin-hao su *, stonehill college...

On k-Edge-magic Halin Graphs

Sin-Min Lee, San Jose State University

Hsin-hao Su*, Stonehill College

Yung-Chin Wang, Tzu-Hui Institute of Technology

41th SICCGTCAt

Florida Atlantic University

March 9, 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

Proof: The sum of all edges is Every edge is counted twice in the vertex

sums.

pkqq mod012

2

1 qkkq

Basic Number Theory

Proposition: Let d = gcd(a,m).

ax = b has a solution in Zm iff d | b.

Moreover, if d | b, then there are exactly d solutions in Zm.

k-Edge-Magic is periodic

Theorem: If a (p,q)-graph G is k-edge-magic then it is pt+k-edge-magic for all t ≥ 0 .

Halin Graphs

Definition: Halin graphs are planar connected graphs that consist of a tree and a cycle connecting the end vertices of the tree.

Wheels

For n > 3, the wheel on n vertices, Wn is a graph with n vertices x1, x2,..., xn, x1 having degree n-1 and all the other vertices having degree 3.

Wheels – Wn

# of vertices: n. # of edges: 2n-2. Necessary condition:

nk

nk

nknn

mod46

mod232

mod1222220

nk mod64

Wheels – Wn‘s Possible k

Necessary condition: Let d = gcd(4,n). For t ≥ 1,

If n=4t, then d = 4. If n=4t+2, then d = 2. If n=4t+1 or 4t+3, then d = 1.

Possible k: If n=4t, then there is no k. If n=4t+2, then k=t+2 or 3t+3 (mod n). If n=4t+1, then k=2t+2 (mod n). If n=4t+3, then k=2t+3 (mod n).

nk mod64

Wheels – not k-EM Theorem: The Halin graph of Wn for

n = 4t is not k-edge-magic for all k.

Wheels – W5

Theorem: The Halin graph of W5 is k-edge-magic for all k ≡ 4 (mod 5).

Wheels – W6

Theorem: The Halin graph of W6 is k-edge-magic for all k ≡ 0,3 (mod 6).

Wheels – W7


Wheels – W9


Wheels – W2n+1

Theorem: The Halin graph of W2n+1 is k-edge-magic for all k ≡ n+2 (mod 2n+1).

Double Stars

Definition: The double star D(m,n) is a tree of diameter three such that there are m appended edges on one ends of P2 and n appended edges on another end.

Double Stars – D(m,n)

# of vertices: m+n+2. # of edges: 2(m+n)+1. Necessary condition:

2mod612

2mod243

2mod1212120

nmk

nmk

nmknmnm

2mod126 nmk

Double Stars – Possible k Necessary condition: Let d = gcd(6,m+n+2). Then, d |12. Possible k: For t ≥ 1,

If m+n=6t-4, then k=2 or 3t+1 (mod m+n+2). If m+n=6t-3, then k=2 (mod m+n+2). If m+n=6t-2, then k=2 or 3t+1 (mod m+n+2). If m+n=6t-1, then k=2,t+2,2t+2,3t+2,4t+2,5t+2

(mod m+n+2). If m+n=6t, then k=2 or 3t+3 (mod m+n+2). If m+n=6t+1, then k=2 or 2t+3 or 4t+4 (mod m+n+2).

2mod126 nmk

Double Stars – D(2,2) Theorem: The Halin graph of D(2,2),

H(D(2,2)), is k-edge-magic for all k.

Double Stars – D(2,2)

Double Stars – D(2,3) Theorem: The Halin graph of D(2,3) is

k-edge-magic for all k ≡ 2 (mod 7).


k-edge-magic for all k ≡ 2,6 (mod 8).

Spiders – Sp(2,0,2) Theorem: The Halin graph of Sp(2,0,2)

is k-edge-magic for all k ≡ 6 (mod 7).


is k-edge-magic for all k ≡ 0,5 (mod 10).

L-Product of Stars with Stars Theorem: The Halin Graph of St(3)xLSt(2)


Spiders – Sp(1n,22) Theorem: The Halin graph of Sp(1n,22)

with n = 2i-5 are not k-edge-magic for all k.

Spiders – Sp(11,22) Theorem: The Halin graph of Sp(11,22)


Order 8 – Not k-EM Theorem: Among 21 Halin graphs of

order 8, the following graphs are not k-edge-magic for all k.

Order 8 – 2-EM and 6-EM Theorem: The following Halin graphs

of order 8 is k-edge-magic for all k ≡ 2,6 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs

of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 7-EM only Theorem: The following Halin graphs of

order 8 is k-edge-magic for k ≡ 7 (mod 8).

Order 8 – 8-EM only Theorem: The following Halin graphs of

order 8 is k-edge-magic for k ≡ 7 (mod 8).

on k-edge-magic halin graphs sin-min lee, san jose state university hsin-hao su *, stonehill college...

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