G-Design of Complete Multipartite Graph Where G Is Five Points-Six Edges

In this paper, we construct G-designs of complete multipartite graph, where G is five points-six edges.


Introduction
Let K v be a complete graph with v vertices, and G be a simple graph with no isolated vertex.A G-design (or G-decomposition) is a pair (X, B), where X is the vertex set of K v and B is a collection of subgraphs of K v , called blocks, such that each block is isomorphic to G and any edge of K v occurs in exactly a blocks of B. For simplicity, such a G-design is denoted by G-GD (v).Obviously, the necessary conditions for the existence of a G-GD(v) are where d is the greatest common divisor of the degrees of the vertices in V(G).
Let be a complete multipartite graph with vertex set , where these X i are disjoint and is a collection of subgraphs of called blocks, such that each block is isomorphic to G and any edge of occurs in exactly a blocks of B. When the multipartite graph has k i partite of size n i 1 ≤ i ≤ r, the holey G-design is denoted by . When (also known as G-decomposition of complete multipartite graph K n (t)).
On the G-design of existence has a lot of research.Let k be the vertex number of G, When k ≤ 4, J. C. Bermond proved that condition (1) is also sufficient in [1]; When k = 5, J. C. Bermond gives a complete solution in [2].When G = S k , P k and C k , K. Ushio investigated the existence of G-design of complete multipartite graph in [3].In this paper, G-designs of complete multipartite graph, where G is five points-six edges is studied.Necessary and sufficient conditions are given for the G-designs of complete multipartite graph K n (t).For graph theoretical term, see [4].

Fundamental Theorem and Some Direct Construction
Let G be a simple graph with five points-six edges (see

Graph 1). G is denoted by (a, b, c)-(c, d, e).
The lexicographic product 1 2 of the graphs G 1 and G 2 is the graph with vertex set V(G 1 ) × V(G 2 ) and an edge joining (u 1 , u 2 ) to (v 1 , v 2 ) if and only if either u 1 is adjacent to v 1 in G 1 or u 1 = v 1 and u 2 and v 2 are adjacent in G 2 .We are only concered with a particular kind of lexicographic product, , Take any  latin square and consider each element in the form   , ,    where  denotes the row,  the column and  the entry, with 1  , , l     .We can construct l 2 graphs G.Copyright © 2012 SciRes.
Let K be a subset of positive integers.A pairwise balanced design (PBD(v, K)) of order v with block sizes from K is a pair (Y , ),where is a finite set (the point set) of cardinality v and B is a family of subsets (blocks) of which satisfy the properties: 2) Every pair of distinct elements of Y occurs in exactly a blocks of .
Let K be a set of positive integers and If there exists a G-HD(t k ) where k  {3, 4, 5, 6, 8}, then there exists a G-HD(t n ) where n ≥ 3.
Proof.Let X be n(n ≥ 3) element set and Z t be a modulo t residual additive group.For K = {3, 4, 5, 6, 8}, take Y = X × Z t , by applying Lemma 2.2, we assume that (X, ) be a PBD(n, k).In the A, we take a block A, for A A k K   , as there exists a G-HD(t k ), let A × Z t be the vertex set of G-HD(t k ) and block set be .

be a , so (Y, ) be a G-HD(tn). A
A Similar to the proof of lemma 2.5, We have the following conclusions.

Graph 1 .
Let G be a simple graph with five points-six edges.