On Z 2 Z 2 -magic Graphs Yihui Wen, Suzhou Science and Technology College Sin-Min Lee, San Jose...
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Transcript of On Z 2 Z 2 -magic Graphs Yihui Wen, Suzhou Science and Technology College Sin-Min Lee, San Jose...
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.
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.
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)).
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.
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.
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
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.
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(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
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(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.