On Z 2  Z 2 -magic Graphs Yihui Wen, Suzhou Science and Technology College Sin-Min Lee, San Jose State University Hsin-hao Su *, Stonehill College 40th SEICCGTC At Florida Atlantic University March 4, 2009

Labelings For any abelian group A written additively we denote A*=A-{0}. Any mapping l:E(G)  A* is called a labeling. Given a labeling on edge set of G we can induced a vertex set labeling l + : V(G)  A as follows: l + (v)=  {l(u,v) : (u,v) in E(G)}

A-magic A graph G is called A-magic if there is a labeling l: E(G)  A* such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant; i.e., l + (v) = c for some fixed c in A. In general, G may admits more than one labeling to become a A-magic graph.

Klein-four group Z 2  Z 2 The Klein-four group V 4 is the direct sum Z 2  Z 2. Its multiplication looks like ⊕ (0,0)(0,1)(1,0)(1,1) (0,0) (0,1)(1,0)(1,1) (0,1) (0,0)(1,1)(1,0) (1,1)(0,0)(0,1) (1,1) (1,0)(0,1)(0,0)

Klein-four group V 4 For simplicity, we denote (0,0), (0,1), (1,0) and (1,1) by 0, a, b and c, respectively. Thus, V 4 is the group {0, a, b, c} with the operation + as: +0abc 00abc aa0cb bbc0a ccba0

Z 2  Z 2 –magic and Z 2 –magic Theorem: The wheel graph W n+1 = N 1 +C n is both Z 2  Z 2 –magic and Z 2 –magic if n is odd.

Odd Graphs Definition: A graph G is called an odd graph if its degree sequence has the property that d i is odd for all i>1. Theorem: All odd graphs are both Z 2  Z 2 – magic and Z 2 –magic.

Example : K 4 K 4 is Z 2  Z 2 –magic with sum 0 and Z 2 – magic with sum 1.

K 4 (a 1,a 2,a 3 ) Definition: A graph K 4 (a 1,a 2,a 3 ) where a i >1 for all i=1,2,3 is a graph which is formed by a K 4 where V(K n )={v 0,v 1,v 2,v 3 } and for each v i, where i=1,2,3, there exists a i pendant edges.

K 4 (a 1,a 2,a 3 ) Corollary: A K 4 (a 1,a 2,a 3 ) where a i >1 for all i=1,2,3 is both Z 2  Z 2 –magic and Z 2 –magic if a i is even for all i=1,2,3.

Z 2  Z 2 –magic but not Z 2 -magic Theorem: The wheel graph W n+1 = N 1 +C n is Z 2  Z 2 –magic, but not Z 2 –magic if n is even.

d(G)= d(G) is Z 2  Z 2 –magic, but not Z 2 –magic.

Amalgamation of copies of d(g) We rename d(g) as (G,u). The amalgamation of copies of (G,u) is formed by gluing the pendent vertex u of each copy into a common vertex. We denote the amalgamation of n copies of (G,u) by Amal(n,(G,u)). Thus, (G,u) is Amal(1,(G,u)).

Amal(n,(G,u)) Theorem: The Amal(n,(G,u)) is Z 2  Z 2 – magic but not Z 2 –magic if n is odd.

Sub(C 2n (∑), §) Let C 2n be a cycle with vertex-set V(C 2n ) ={c 1,…,c n, c 1 *,…,c n *} and ∑ be a permutation of {1,2,…,n}. We construct a cubic graph C 2n (∑) as follows: V(C 2n (∑)) = V(C 2n ) E(C 2n (∑)) = E(C 2n )  { ( c i,c ∑(i)* ): i =1,2,…,n} Now let § : E(C 2n )→N be a mapping. If §(e i ) = a i and a i not equal 0, then we subdivide the edge e i by insert a i new vertices in the edge e i. If §(e i ) = 0, we do not add any vertex in e i.

Example

Sub(C 2n (∑), §) Theorem: For any n > 2, and any ∑, §, the subdivision graph Sub(C 2n (∑),§) is Z 2  Z 2 – magic. It is not Z 2 –magic if § is not a zero mapping.

Comb(n) Definition: A graph Comb(n) where n is a positive integer which is greater than or equal to 3 is a graph formed by a path P n and for each inside vertices v i, where i=2,3,…,n-1, there exists a pendant edge with a vertex. We call these pendant vertices by v i *.

Village(n, F) Definition: Let P n be the path within a graph Comb(n). Let F: E(P n )→N be a mapping. A graph Village(n,F) is a graph which is formed by a Comb(n) and glue a path P F(ei)+2 to v i * and v i+1 * for all i=2,…,n- 2 or to v i and v i+1 * for i=1 or n-1.

Examples

Village(n, F) Theorem: For any n > 3, and any Comb(n), and any F, the graph Village(n, F) is Z 2  Z 2 –magic, but not Z 2 –magic.

Village(n) Definition: Let P n and P n * be two paths of length n>2 where their vertices are named by v 1,v 2,…,v n and v 1 *,v 2 *,…,v n *, respectively. A graph Village(n) is a graph which is formed by P n and P n * with extra vertices r 1,r 2,…,r n-1 and extra edges (v i,v i *) for all i=1,…,n and (r i,v i *) and (r i, v i+1 *) for all i=1,2,…,n-1

Village(n), n=2,3

Village(n), n=4,5

Not Z 2  Z 2 –magic but Z k -magic Theorem: The Amal(n,(G,u)) is not Z 2  Z 2 –magic but Z 3 – magic if n is even

Pagoda(1) Definition: Pagoda(1) is a graph which combines one edge of 3-cycle C 3 with one edge side of 4- cycle C 4. Theorem (Chou and Lee): Pagoda(1) is is not Z 3 –magic.

Pagoda(n) Definition: Pagoda(2) is a graph which combines the bottom edge of Pagoda(1) with one edge of 4-cycle C 4. Pagoda(n) is a graph which combines the bottom edge of Pagoda (n-1) with one edge of 4-cycle C 4.

Pagoda(n) Theorem (Chou and Lee): Pagoda(n) is Z 3 –magic for all n > 1. Theorem: Pagoda(n) is Z 2  Z 2 –magic for all n.

Pagoda(n)

Mongolian Tent Definition: Mongolian Tent (1), or MT (1), is Pagoda (1). Definition: MT(2) is a graph which combines vertices and edges of the right hand side of a MT(1) with vertices and edges of the left hand side of another MT(1).

Mongolian Tent MT(n) Definition: MT(n) is a graph which combines vertices and edges of the right hand side of a MT(n-1) with vertices and edges of the left hand side of another MT(1), the corresponding vertices and edges are similar to the construction of MT(2).

MT(2) Theorem (Chou and Lee): Mongolian Tent MT(n) is not Z 3 –magic for n =1, 2.

MT(n) Theorem (Chou and Lee): Mongolian Tent MT(n) is Z 3 –magic for all n > 2.

MT(n) Theorem: Mongolian Tent MT(n) is Z 2  Z 2 –magic for all n.

Womb Graphs Definition: A womb μ(n;a 1,…,a n ) where n>3 is a unicyclic graph which is formed by a cycle C n where V(C n )={v 1,v 2,…,v n } and for each v i, there exists a i pendant edges.

μ(3;1,3,5) μ(3;1,3,5) is both Z 2  Z 2 –magic and Z 2 –magic.

μ(n;a 1,…,a n ), a i are odd Theorem: The graph μ(n;a 1,…,a n ) is Z 2  Z 2 –magic and Z 2 –magic if a i is odd and greater or equal to 1 for all i=1,2,…,n.

μ(n;a 1,…,a n ), a i are not all odd Theorem: The graph μ(n;a 1,…,a n ) is Z 2  Z 2 –magic, for any n  3 and a 1,…,a n are not all zero if and only if the number of the vertices in the cycle with even number of pendants is even (i.e. the number of even numbers of a 1,…,a n is even.)