Download presentation

Presentation is loading. Please wait.

Published byDominick Plock Modified over 3 years ago

1
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

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 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)

5
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

6
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.

7
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.

8
Example : K 4 K 4 is Z 2 Z 2 –magic with sum 0 and Z 2 – magic with sum 1.

9
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.

10
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.

11
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.

12
d(G)= d(G) is Z 2 Z 2 –magic, but not Z 2 –magic.

13
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)).

14
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.

15
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.

16
Example

17
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.

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 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 *.

19
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.

20
Examples

21
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.

22
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

23
Village(n), n=2,3

24
Village(n), n=4,5

25
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

26
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.

27
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.

28
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.

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 Z 3 –magic for n =1, 2.

33
MT(n) Theorem (Chou and Lee): Mongolian Tent MT(n) is Z 3 –magic for all n > 2.

34
MT(n) Theorem: Mongolian Tent MT(n) is Z 2 Z 2 –magic for all n.

35
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.

36
μ(3;1,3,5) μ(3;1,3,5) is both Z 2 Z 2 –magic and Z 2 –magic.

37
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.

38
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.)

Similar presentations

Presentation is loading. Please wait....

OK

Graphs 9.1 Graphs and Graph Models أ. زينب آل كاظم 1.

Graphs 9.1 Graphs and Graph Models أ. زينب آل كاظم 1.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google