Some results on the injective chromatic number of graphs

Abstract

A k-coloring of a graph G=(V,E) is a mapping c:V?{1,2,?,k}. The coloring c is injective if, for every vertex v?V, all the neighbors of v are assigned with distinct colors. The injective chromatic number ? i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K 4-minor free graph G with maximum degree Δ?1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$ . Moreover, some related results and open problems are given.