:In this paper, the vertex partition number of a claw-free graph is studied, the formula for vertex partition of a complete n-partition graph will be obtained, finally,it is proved that the vertex linear arboricity of a graph is not greater than its edge linear arboricity and the inequality is sharp.

This paper constructs the edge clique cover of minimum size and give the edge clique cover number for the complete pentapartite graphs Kn,n,n,n,n where n≡1,5(mod6).

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.