on mod( k )-edge -magic cubic graphs
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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 MCCCCAt
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)-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.
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
Since q is an integer, p must be even.
2
3pq
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 n-cycle 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 2-regular subgraph with 3p/4 or 3p/4 edges.