G-design of λ-fold complete multipartite graph where G is three kinds of graphs with five points;

The existence of G-design ofλ-fold complete multipartite graph is discussed where G is the 3-path and a stick necessary and sufficient conditions are given for the G-design ofλKn(t).

Let λKn(g)be aλ-fold complete multipartite graph and G be a finite simple graph．A(λKn(g),G)-design is a partition of the edges ofλKn(g)into sub-graphs each of which is isomorphic to G．In this paper the existence of a G-design ofλKn (g)was discussed where G is the 4-cycle and a stick．Necessary and sufficient conditions were given for theλKn(g)-design

K1,pq - factorization of complete bipartite multigraphs;

LetλK_(m,n) be a complete bipartite multigraph with two partite sets having m and n vertices, respectively.

On property M(5) of some complete multipartite graphs;

Mandatory decomposition of complete multipartite graph into cycles of lengths 3,4 and 5;

It is easy to see that Ohba s conjecture is true if and only if it is true for complete multipartite graphs.

The concept Mandatory Decomposition is introduced in the paper, proving the existence of{C 3, C 4}- and{C 3, C 6}- mandatory decompositions of complete multipartite graphs K r(t).

In this paper,it is shown that a necessary and sufficient condition for the existence of a p2k + 1 - factorization of the balanced bipartite multigraphsλK n,n is n ≡ 0 (mod 4k(2k + 1)/d), where d = gcd(λ,4k)≠1.