Each vertex of a graph G = （V, E） is said to dominate every vertex in its closed neighborhood. A set S C V is a double dominating set for G if each vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double dominating number dd（G）. In this paper, new relationships between dd（G） and other domination parameters are explored and some results of [1] are extended. Furthermore, we give the Nordhaus-Gaddum-type results for double dominating number.
A proper k-total coloring f of the graph G（V, E） is said to be a k-vertex strong total coloring if and only if for every v ∈ V（G）, the elements in N[v] are colored with different colors, where N[v] =. {u｜uv E V（G）} ∪{v}. The value xT^vs（G） = min{k｜ there is a k-vertex strong total coloring of G} is called the vertex strong total chromatic number of G. For a 3-connected plane graph G（V, E）, if the graph obtained from G（V, E） by deleting all the edges on the boundary of a face f0 is a tree, then G（V, E） is called a Halin-graph. In this paper, xT^vs,8（G） of the Halin-graph G（V,E） with A（G） 〉 6 and some special graphs are obtained. Furthermore, a conjecture is initialized as follows： Let G（V, E） be a graph with the order of each component are at least 6, then xT^vs（G） ≤ △（G） ＋ 2, where A（G） is the maximum degree of G.
