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 Z p.

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 Z k.

Examples A Mod(k)-EM graph for k = 2,3,4,6, but not a Mod(5)-EM graph.

Examples The path P 4 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.

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 C n xP 2 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 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.cubiccirculant grapheven number cycle

Möbius Ladders A möbius ladder ML(2n) with the vertices denoted by a 1, a 2, …, a 2n. The edges are then {a 1, a 2 }, {a 2, a 3 }, … {a 2n, a 1 }, {a 1, a n+1 }, {a 2, a n+2 }, …, {a n, a 2n }.

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 {v i, u i : 0 ≤ i ≤ n-1} and edges {v i v i+1, v i u i, u i u i+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 C 2n with vertices a 1, a 2, …, a n, b 1, b 2, …, b n such that a 1 is adjacent to b 1, and a i is adjacent to b n+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 = { x j, c i,j : i =1,2,3, j = 1, 2, …, n} such that c i,1, c i,2, …, c i,n are three disjoint cycles and x j is adjacent to c 1,j, c 2,j, c 3,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 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 {a 1, a 2, a 3, a 4, b 3, b 2 }, {a 5, a 6, a 7, a 8, b 7, b 6 }, …, {a 4k-3, a 4k-2, a 4k-1, a 4k, b 4k-1, b 4k-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 C n with a star St(3), and then join all the leaves of the stars by a cycle C 2n. 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.