On Mod(k)-Edge-magic Cubic
Graphs
Sin-Min Lee, San Jose State University
Hsin-hao Su*, Stonehill College
Yung-Chin Wang, Tzu-Hui Institute of Technology
24th MCCCC
At
Illinois State University
September 11, 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.
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.
Problem

Chopra, Kwong and Lee in 2006
proposed a problem to characterize
Mod(2)-EM 3-regular graphs.
Cubic Graphs


Definition: 3-regular (p,q)-graph is
called a cubic graph.
The relationship between p and q is
q
3p
2

Since q is an integer, p must be even.
One for All


Theorem: If a cubic graph is Mod(k)edge-magic with vertex sum s (mod k),
then it is Mod(k)-edge-magic for all other
vertex sum s as long as gcd(k,3)=1.
Proof:

Since every vertex is of degree 3, by adding
or subtracting 1 to each adjacent edge, the
vertex sum increases by 1. Since gcd(k,3)=1,
it generates all.
Sufficient Condition


Theorem: If a cubic graph G of order p
has a 2-regular subgraph with length
3p/4 or 3p/4, then it is Mod(2)-EM.
Proof:


Note that since G is a cubic graph, p is even.
We provide two lebelings for each p = 4s or
4s+2.
When p = 4s

Two Labelings:


Label the edges of the cycle either by even
numbers, 2, 4, ..., 6s. The remaining 3s edges
are labeled by 1, 3, 5, ..., 6s-1.
Label the edges of the cycle either by odd
numbers, 1, 3, 5, ..., 6s-1. The remaining 3s
edges are labeled by even numbers 2, 4, ..., 6s.
Examples
When p = 4s + 2

Two Labelings:


If G has a cycle with length 3p/4. Label the
edges of the cycle 3s+1 by even numbers, 2, 4,
..., 6s, 6s+2. The remaining 3s+2 edges are
labeled by 1, 3, 5, ..., 6s+1,6s+3 .
If G has a cycle with length 3p/4. Label the
edges of the cycle 3s+2 by odd numbers, 1, 3,
5, ..., 6s+3.. The remaining 3s+1 edges are
labeled by even numbers 2, 4, ..., 6s+2.
Examples
Cylinder Graphs

Theorem: A cylinder graph CnxP2 is
Mod(2)-EM if n ≠ 2 (mod 4) for 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(2)-EM for all n ≥ 3.
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.
Theorem: The generalized Petersen graph
P(n,t) is a Mod(2)-EM graph for all k ≥ 3 if
n is odd.
Gen. Petersen Graph Ex.
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(2)-EM for all n ≥ 3.
Turtle Shell Graphs Examples
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(2)-EM for all n ≥ 3.
Issacs Graphs Examples
Twisted Cylinder Graphs


Theorem: A twisted cylinder graph TW(n)
is Mod(2)-EM if n ≠ 2 (mod 4).
Proof:


If n  2 (mod 4), say n = 4k+2 then the graph
TW(n) has order 8k+4 and size 6(2k+1).
If it is Mod(2)-EM then it has a 2-regular
subgraph with length 3(2k+1). As TW(n) is
bipartite, it is impossible.
Proof (continued)

Proof:



If n  0 (mod 4), say n = 4k, then the graph
TW(n) has order 8k and size 12k.
We want to show it has a 2-regular
subgraph with length 6k.
Label k disjoint 6-cycles {a1, a2, a3, a4, b3,
b2}, {a5, a6, a7, a8, b7, b6}, …, {a4k-3, a4k-2,
a4k-1, a4k, b4k-1, b4k-2} by even numbers and
all the remaining edges by odd numbers.
Twisted Cylinder Graphs Ex.
Tutte Graphs


For any complete binary graph B(2,k), k
> 1, we append an edge on the root then
hang off of each leaf a 2t+1-cycle (t > 2)
with t independent chords not incident to
the leaf.
We denote this cubic graph by
Tutte(B(2,k), t).
Tutte Graphs



The cubic graph with longest cycle
length 2t+1.
For it is inspired by Tutte’s construction
of Tutte(B(2,1), 2).
Theorem: The Tutte(B(2,k),t) is
Mod(2)-EM for all k,t ≥ 1.
Tutte Graph Examples
Sufficient Condition Extended


Theorem: If a cubic graph G of order p
has a 2-regular subgraph with 3p/4 or
3p/4 edges, then it is Mod(2)-EM.
Proof:

The same labelings work here.
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(2)-EM for all n ≥ 3.
Coxeter Graph Examples
Necessary Condition


Theorem: If a cubic graph G of order p is
Mod(2)-EM, then it has a 2-regular
subgraph with 3p/4 or 3p/4 edges.
Proof:


As a cubic graph, p must be even.
Since G has 3p/2 edges, it has either 3p/4
odd and 3p/4 even edges or 3p/4 odd and
3p/4 even edges.
Proof (continued)

Proof:



Since gcd(2,3)=1, if G is Mod(2)-EM with
sum 0, then it is Mod(2)-EM with sum 1.
Assume that G is Mod(2)-EM with sum 0.
With vertex sum equals 0, there are only two
possible labelings:
Proof (continued)

Proof:
Proof (continued)

Proof:



Pick an odd edge. Then there must be another
odd edge attached to its vertex.
Keep traveling through odd edges.
Since there is always another odd edge to
travel through, you stop only when you reach
the initial odd edge.
Classification

Theorem: If a cubic graph G of order p
is Mod(2)-EM if and only if it has a 2regular subgraph with 3p/4 or 3p/4
edges.