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A star edge-coloring of G is a proper edge coloring of G such that any path of length4in G is not bicolored, and a star total coloring of G is a proper total coloring of G such that the vertices and edges of any path of the length of two have received distinct colors. In this paper, we mainly deal with the star edge-coloring and star total coloring of some composition graphs. Based on the graph decomposition research, we give an upper bound on star edge-coloring chromatic index of some composition graphs, and obtain the chromatic index of some composition graphs.Firstly, we give an upper bound on star edge-coloring chromatic index of the complete balanced bipartite graph: Then we improved the upper bound of the star edge-coloring chromatic index of the join graph Pn v Pm, even more, we get the following upper bounds on star edge-coloring chromatic index of some composition graphs:(1) Let G be a simple graph of order n and H be a simple graph of order m, then(2) Let G and H are both paths, thenSecondly, we studied the star total coloring of some composition graphs. We use the means of graph decomposition and the method of constructing to determine the star total chromatic number of the composition graph, and the results as follows:(1) Let G be a wheel of order n+1, or fan, or star. if H is a simple graph of order in such that the star total chromatic number is m with no vertexes oven the same coloring, then if H is a simple graph of order m such that χ’(H)=Δ(H)=m-1, m≥5, then (2) Let G be a complete graph Km, if H is a simple graph of order m such that the star total chromatic number is m with no vertexes oven the same coloring, then if H is a simple graph of order m such that x’(H)=ΔA(H)=m-1, and mn is even, then(3) Let G be a circle Cm, if H=Kmc,then if H=Ω, and m≥4,n≥8,then
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