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

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.
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.
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.
Examples: Edge-Magic

The following maximal outerplanar
graphs with 6 vertices are EM.
Examples: Edge-Magic

In general, G may admits more than
one labeling to become an edge-magic
graph with different vertex sums.
Mod(k)-Edge-Magic Graphs


Let k ≥ 2.
A (p,q)-graph G is called Mod(k)-edgemagic (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.
Examples

A Mod(k)-EM graph for k = 2,3,4,6, but
not a Mod(5)-EM graph.
Examples

The path P4 with 4 vertices is Mod(2)-EM,
but not Mod(k)-EM for k = 3,4.
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.
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.
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.
Cubic Graphs

Definition: 3-regular (p,q)-graph is
called a cubic graph.
The relationship between p and q is
3p
q
2

Since q is an integer, p must be even.

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.
Examples
Cylinder Graphs

Theorem: A cylinder graph CnxP2 is
Mod(3)-EM for all n ≥ 3.
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 ncycle by adding edges (called “rungs”)
connecting opposite pairs of vertices in the
cycle.
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}.
Möbius Ladders

Theorem: A Möbius ladder ML(2n) is
Mod(3)-EM for all even n ≥ 4.
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.
Turtle Shell Graphs Examples
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.
Coxeter Graph Examples
Corollaries


Corollary: If a cubic graph is
Hamiltonian, then it is Mod(3)-EM.
Corollary: Almost all cubic graphs are
Mod(3)-EM.
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).
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.
Issacs Graph’s Inner Cycle
Issacs Graphs Examples
Twisted Cylinder Graphs


Theorem: All twisted cylinder graph
TW(n) are Mod(3)-EM.
Remark: Twisted cylinder graph TW(n) is
NOT hamiltonian.
Twisted Cylinder Graphs Ex.
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.
[2] S-M. Lee, W.M. Pigg, T.J. Cox, On Edge-Magic Cubic Graphs Conjecture,
Congressus Numeratium 105 (1994), 214-222.
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.
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.
Degree 3 Vertices
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.
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).
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 ).
Petersen Graph Example
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.
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?