On Z 2  Z 2 -magic Graphs

1. On Z2Z2-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

2. 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)}

3. 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.

4. Klein-four group Z2Z2 • The Klein-four group V4 is the direct sum Z2Z2. Its multiplication looks like

5. 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:

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

7. 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 Z2Z2–magic and Z2–magic.

8. Example : K4 • K4 is Z2Z2–magic with sum 0 and Z2–magic with sum 1.

9. 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.

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

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

12. d(G) is Z2Z2–magic, but not Z2–magic. d(G)=<1,2,3>

13. 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)). Amalgamation of copies of d(g)

14. Amal(n,(G,u)) Theorem:The Amal(n,(G,u)) is Z2Z2–magic but not Z2–magic if n is odd.

15. 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.

16. Example

17. Sub(C2n (∑), §) Theorem:For any n > 2, and any ∑ , §, the subdivision graph Sub(C2n (∑),§) is Z2Z2–magic. It is not Z2–magic if § is not a zero mapping.

18. 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*.

19. 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.

20. Examples

21. Village(n, F) Theorem:For any n > 3, and any Comb(n) , and any F, the graph Village(n, F) is Z2Z2–magic, but not Z2–magic.

22. 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

23. Village(n), n=2,3

24. Village(n), n=4,5

25. Not Z2Z2–magic but Zk-magic Theorem:The Amal(n,(G,u)) is not Z2Z2–magic but Z3–magic if n is even

26. 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.

27. 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.

28. Pagoda(n) Theorem (Chou and Lee):Pagoda(n) is Z3–magic for all n > 1. Theorem:Pagoda(n) is Z2Z2–magic for all n.

29. Pagoda(n)

30. 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).

31. 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).

32. MT(2) Theorem (Chou and Lee):Mongolian Tent MT(n) is not Z3–magic for n =1, 2.

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

34. MT(n) Theorem:Mongolian Tent MT(n) is Z2Z2–magic for all n.

35. 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.

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

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

38. μ(n;a1,…,an), ai are not all odd Theorem:The graph μ(n;a1,…,an) is Z2Z2–magic, for any n3 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.)