On Z2Z2-magic Graphs

Yihui Wen, Suzhou Science and Technology College

Sin-Min Lee, San Jose State University

Hsin-hao Su*, Stonehill College

40th SEICCGTCAt

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 Z2Z2

The Klein-four group V4 is the direct sum Z2Z2. Its multiplication looks like

⊕ (0,0) (0,1) (1,0) (1,1)

(0,0) (0,0) (0,1) (1,0) (1,1)

(0,1) (0,1) (0,0) (1,1) (1,0)

(1,0) (1,0) (1,1) (0,0) (0,1)

(1,1) (1,1) (1,0) (0,1) (0,0)

Klein-four group V4

For simplicity, we denote (0,0), (0,1), (1,0) and (1,1) by 0, a, b and c, respectively. Thus, V4 is the group {0, a, b, c} with the operation + as:

+ 0 a b c

0 0 a b c

a a 0 c b

b b c 0 a

c c b a 0

Z2Z2–magic and Z2–magic

Theorem: The wheel graph Wn+1 = N1+Cn is both Z2Z2–magic and Z2–magic if n is odd.

Odd Graphs

Definition: A graph G is called an odd graph if its degree sequence <d1,d2,…, dn> has the property that di is odd for all i>1 .

Theorem: All odd graphs are both Z2Z2–magic and Z2–magic.

Example : K4

K4 is Z2Z2–magic with sum 0 and Z2–magic with sum 1.

K4(a1,a2,a3)

Definition: A graph K4(a1,a2,a3) where ai >1 for all i=1,2,3 is a graph which is formed by a K4 where V(Kn)={v0,v1,v2,v3} and for each vi, where i=1,2,3, there exists ai pendant edges.

K4(a1,a2,a3)

Corollary: A K4(a1,a2,a3) where ai >1 for all i=1,2,3 is both Z2Z2–magic and Z2–magic if ai is even for all i=1,2,3.

Z2Z2–magic but not Z2-magic

Theorem: The wheel graph Wn+1 = N1+Cn is Z2Z2–magic, but not Z2–magic if n is even.

d(G)=<1,2,3>

d(G) is Z2Z2–magic, but not Z2–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 Z2Z2–magic but not Z2–magic if n is odd.

Sub(C2n (∑), §)

Let C2n be a cycle with vertex-set V(C2n) ={c1,…,cn, c1*,…,cn*} and ∑ be a permutation of {1,2,…,n}. We construct a cubic graph C2n (∑) as follows:

V(C2n (∑)) = V(C2n )

E(C2n (∑)) = E(C2n ) { ( ci,c∑(i)* ): i =1,2,…,n}

Now let § : E(C2n )→N be a mapping. If §(ei) = ai and ai not equal 0, then we subdivide the edge ei by insert ai new vertices in the edge ei.

If §(ei) = 0 , we do not add any vertex in ei.

Example

Sub(C2n (∑), §)

Theorem: For any n > 2, and any ∑ , §, the subdivision graph Sub(C2n (∑),§) is Z2Z2–magic. It is not Z2–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 Pn and for each inside vertices vi, where i=2,3,…,n-1, there exists a pendant edge with a vertex. We call these pendant vertices by vi*.

Village(n, F)

Definition: Let Pn be the path within a graph Comb(n). Let F: E(Pn)→N be a mapping. A graph Village(n,F) is a graph which is formed by a Comb(n) and glue a path PF(ei)+2 to vi* and vi+1* for all i=2,…,n-2 or to vi and vi+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 Z2Z2–magic, but not Z2–magic.

Village(n)

Definition: Let Pn and Pn* be two paths of length n>2 where their vertices are named by v1,v2,…,vn and v1*,v2*,…,vn*, respectively. A graph Village(n) is a graph which is formed by Pn and Pn* with extra vertices r1,r2,…,rn-1 and extra edges (vi,vi*) for all i=1,…,n and (ri,vi*) and (ri, vi+1*) for all i=1,2,…,n-1

Village(n), n=2,3

Village(n), n=4,5

Not Z2Z2–magic but Zk-magic

Theorem: The Amal(n,(G,u)) is not Z2Z2–magic but Z3–magic if n is even

Pagoda(1)

Definition: Pagoda(1) is a graph which combines one edge of 3-cycle C3 with one edge side of 4-cycle C4.

Theorem (Chou and Lee): Pagoda(1) is is not Z3–magic.

Pagoda(n)

Definition: Pagoda(2) is a graph which combines the bottom edge of Pagoda(1) with one edge of 4-cycle C4.

Pagoda(n) is a graph which combines the bottom edge of Pagoda (n-1) with one edge of 4-cycle C4.

Pagoda(n)

Theorem (Chou and Lee): Pagoda(n) is Z3–magic for all n > 1.

Theorem: Pagoda(n) is Z2Z2–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 Z3–magic for n =1, 2.

MT(n)

Theorem (Chou and Lee): Mongolian Tent MT(n) is Z3–magic for all n > 2.

0

00 0 0 0 0 0

0 0 0 0 0 0

011 1 11

1 1111

111 1 1

2

2

2 2

2

2 2 2

2 2 2 2

1

MT(n)

Theorem: Mongolian Tent MT(n) is Z2Z2–magic for all n.

Womb Graphs

Definition: A womb μ(n;a1,…,an) where n>3 is a unicyclic graph which is formed by a cycle Cn where V(Cn)={v1,v2,…,vn} and for each vi, there exists ai pendant edges.

μ(3;1,3,5)

μ(3;1,3,5) is both Z2Z2–magic and Z2–magic.

μ(n;a1,…,an), ai are odd

Theorem: The graph μ(n;a1,…,an) is Z2Z2–magic and Z2–magic if ai is odd and greater or equal to 1 for all i=1,2,…,n.

μ(n;a1,…,an), ai are not all odd

Theorem: The graph μ(n;a1,…,an) is Z2Z2–magic, for any n3 and a1,…,an 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 a1,…,an is even.)