on mod(3)-edge -magic graphs
DESCRIPTION
On Mod(3)-Edge -magic Graphs. Sin-Min Lee , San Jose State University Karl Schaffer , De Anza College Hsin-hao Su * , Stonehill College Yung-Chin Wang , Tzu- Hui Institute of Technology 6th IWOGL 2010 At University of Minnesota, Duluth October 22, 2010. Supermagic Graphs. - PowerPoint PPT PresentationTRANSCRIPT
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On Mod(3)-Edge-magic Graphs
Sin-Min Lee, San Jose State University
Karl Schaffer, De Anza College
Hsin-hao Su*, Stonehill College
Yung-Chin Wang, Tzu-Hui Institute of Technology
6th IWOGL 2010At
University of Minnesota, Duluth
October 22, 2010
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Supermagic Graphs For a (p,q)-graph, in 1966, Stewart[1]
defined that a graph labeling is supermagic iff the edges are labeled 1,2,3,…,q so that the vertex sums are a constant.
[1] B.M. Stewart, Magic Graphs, Canadian Journal of Mathematics 18 (1966), 1031-1059.
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Magic Square The classical concept of a magic square of
n2 boxes corresponds to the fact that the complete bipartite graph K(n,n) is super magic if n ≥ 3.
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Edge-Magic Graphs Lee, Seah and Tan in 1992 defined that a
(p,q)-graph G is called edge-magic (in short EM) if there is an edge labeling l: E(G) {1,2,…,q} 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.
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Examples: Edge-Magic The following maximal outerplanar
graphs with 6 vertices are EM.
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Examples: Edge-Magic In general, G may admits more than
one labeling to become an edge-magic graph with different vertex sums.
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Mod(k)-Edge-Magic Graphs Let k ≥ 2. A (p,q)-graph G is called Mod(k)-edge-
magic (in short Mod(k)-EM) if there is an edge labeling l: E(G) {1,2,…,q} 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 k; i.e., l+(v) = c for some fixed c in Zk.
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Examples A Mod(k)-EM graph for k = 2,3,4,6, but
not a Mod(5)-EM graph.
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Examples The path P4 with 4 vertices is Mod(2)-EM,
but not Mod(k)-EM for k = 3,4.
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Paths Theorem: A path P2 is Mod(k)-EM for
all k. Proof: There is only one edge. Must be
labeled 1. Theorem: When n > 2, the path Pn is
Mod(k)-EM if and only if k = 2 and n is even.
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Notations For n > 2, let the vertices of Pn be v1, v2,
v3, …, vn, where v1 and vn are the end vertices of degree 1, and vi is adjacent to vi+1, for i = 1, 2, …, n-1.
Let the edge joining vertices vi and vi+1 be ei, for i = 1, 2, …, n-1.
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Proof Suppose e1 receives edge label m. Then
the vertex v1 is labeled m. For the vertex v2 to be labeled m as
well, edge e2 needs to be labeled 0. Similarly, the remaining edges need to
be labeled by m and 0, alternately. This is only possible when k = 2 and n
is even, in which each vertex labeled 1.
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Cubic Graphs Definition: 3-regular (p,q)-graph is
called a cubic graph. The relationship between p and q is
Since q is an integer, p must be even.2
3pq
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Sufficient Condition Theorem: If a cubic graph G is
Hamiltonian, then it is Mod(3)-EM. Proof:
Note that since G is a cubic graph, p is even. We label all the edges of the cycle by 1, -1
(mod 3) alternatively and the rest edges by 0 (mod 3). It is easy to check that the vertices will be labeled by 0. pkqq mod012
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Examples
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Cylinder Graphs Theorem: A cylinder graph CnxP2 is
Mod(3)-EM for all n ≥ 3.
pkqq mod012
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Möbius Ladders The concept of Möbius ladder was
introduced by Guy and Harry in 1967. It is a cubic circulant graph with an
even number n of vertices, formed from an n-cycle by adding edges (called “rungs”) connecting opposite pairs of vertices in the cycle. pkqq mod012
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Möbius Ladders A möbius ladder ML(2n)
with the vertices denoted by a1, a2, …, a2n. The edges are then {a1, a2}, {a2, a3}, … {a2n, a1}, {a1, an+1}, {a2, an+2}, … , {an, a2n}. pkqq mod012
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Möbius Ladders Theorem: A Möbius ladder ML(2n) is
Mod(3)-EM for all even n ≥ 4.
pkqq mod012
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Turtle Shell Graphs Add edges to a cycle C2n with vertices
a1, a2, …, an, b1, b2, …, bn such that a1 is adjacent to b1, and ai is adjacent to bn+2-i, for i = 2, …, n. The resulting cubic graph is called the turtle shell graph of order 2n, denoted by TS(2n).
Theorem: The turtle shell graph TS(2n) is Mod(3)-EM for all n ≥ 3.
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Turtle Shell Graphs Examples
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Coxeter Graphs For n > 3, we append on each vertex of Cn
with a star St(3), and then join all the leaves of the stars by a cycle C2n. We denote the resulting cubic graph by Cox(n).
Note Cox(n) has 4n vertices. Theorem: The Coxeter graph Cox(n) is
Mod(3)-EM for all n ≥ 3.
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Coxeter Graph Examples
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Corollaries Corollary: If a cubic graph is
Hamiltonian, then it is Mod(3)-EM. Corollary: Almost all cubic graphs are
Mod(3)-EM. pkqq mod012
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Issacs Graphs For n > 3, we denote the graph with
vertex set V = { xj, ci,j: i =1,2,3, j = 1, 2, …, n} such that ci,1, ci,2, …, ci,n are three disjoint cycles and xj is adjacent to c1,j, c2,j, c3,j.
We call this graph Issacs graph and denote by IS(n).
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Issacs Graphs Issacs graphs were first considered by
Issacs in 1975 and investigated in Seymour in 1979.
They are cubic graphs with perfect matching.
Theorem: The Issacs graph IS(2n) is Mod(3)-EM for an even n ≥ 4.
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Issacs Graph’s Inner Cycle
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Issacs Graphs Examples
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Twisted Cylinder Graphs Theorem: All twisted cylinder graph
TW(n) are Mod(3)-EM. Remark: Twisted cylinder graph TW(n) is
NOT hamiltonian. pkqq mod012
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Twisted Cylinder Graphs Ex.
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Conjecture Conjecture[2]: A cubic graph with order p
= 4s+2 is Mod(3)-EM. With the previous examples, this is a
reasonable extension of a conjecture by Lee, Pigg, Cox in 1994. pkqq mod012
[2] S-M. Lee, W.M. Pigg, T.J. Cox, On Edge-Magic Cubic Graphs Conjecture, Congressus Numeratium 105 (1994), 214-222.
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Sufficient Condition Extended Theorem: If a cubic graph G of order p
has a 2-regular subgraph with p edges, then it is Mod(3)-EM.
Proof: The same labelings work here. pkqq mod012
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Mod(2)-EM Classification (Lee, Su, Wang) Theorem: If a cubic
graph G of order p is Mod(2)-EM if and only if it has a 2-regular subgraph with 3p/4 or 3p/4 edges.
Actually, this theorem looks true for all n-regular graphs. The same proof of cubic graphs should apply to n-regular graphs with some minor modifications.
pkqq mod012
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Degree 3 Vertices
pkqq mod012
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Necessary Condition Question: If a cubic graph G of order p is
Mod(3)-EM, then it has a 2-regular subgraph with p edges. pkqq mod012
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Generalized Petersen Graphs The generalized Petersen graphs P(n,k) were
first studied by Bannai and Coxeter. P(n,k) is the graph with vertices {vi, ui : 0 ≤ i
≤ n-1} and edges {vivi+1, viui, uiui+k}, where subscripts modulo n and k.
(Alspach 1983; Holton and Sheehan 1993) The generalized Petersen graph GP(n,k) is nonhamiltonian iff k = 2 and n ≡ 5 (mod 6).
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Generalized Petersen Graphs Theorem: A generalized Petersen graphs
GP(n,k) is Mod(3)-EM for all (n,k) not of the form ( 5 mod(6) , 2 ). pkqq mod012
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Petersen Graph Example
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Necessary Condition Failed The Peterson graph shows that the
necessary condition is not held since it does not have a path of order 10, but it is a Mod(3)-EM. pkqq mod012
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Future Study Is it possible to find an if and only if
condition to classify Mod(3)-EM cubic graphs?
Can we extend the sufficient condition to n-regular graphs? pkqq mod012