The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) ? 2 3 |V (G)|+ 23 6 , where d(G) denotes the degree of a vertex in G, then χT (G) ? d(G) + 2.
A star coloring of an undirected graph G is a proper coloring of G such that no path of length 3 in G is bicolored. The star chromatic number of an undirected graph G, denoted by xs（G）, is the smallest integer k for which G admits a star coloring with k colors. In this paper, we show that if G is a graph with maximum degree A, then xs（G） ≤ [7△3/2]], which gets better bound than those of Fertin, Raspaud and Reed.
