On the vertex-distinguishing proper edge coloring of composition of complete graph and star

A proper k-edge coloring of a simple graph G is called k-vertex-distinguishing proper edge coloring (k-VDPEC) if for any two distinct vertices u and v of G, the set of colors assigned to edges incident to u differs from the set of colors assigned to edges incident to v. The minimum number of colors required for a vertex-distinguishing proper edge coloring of G, denoted by χs′(G), is called the vertex-distinguishing proper edge chromatic number. For p⩾2 and q⩾4, we will obtain vertex-distinguishing proper edge chromatic number of composition of complete graph Kp with order p and star Sq with order q, which is pq. •Coloring problem in graph theory research has important theoretical significance and application value.•The vertex-distinguishing proper edge coloring is interesting and difficult problem.•Determining chromatic number of various kinds of colorings is a fundamental problem of graph coloring.•For p⩾2 and q⩾4, we will obtain vertex-distinguishing proper edge chromatic numbers of Kp[Sq] in this paper.