Note on the perfect EIC-graphs

Three edges e1, e2 and e3 in a graph G   are consecutive if they form a path (in this order) or a cycle of length 3. The injective edge coloring number χi′(G) is the minimum number of colors permitted in a coloring of the edges of G such that if e1, e2 and e3 are consecutive edges in G, then e1 and e3 receive the different colors. Let ω′ denote the number of edges in a maximum clique of G. A graph G is called an ω  ′ edge injective colorable (or perfect EIC-)graph if χi′(G)=ω′. In this paper, we give a sharp bound of the injective coloring number of a 2-connected graph with some forbidden conditions, and then we also characterize some perfect EIC-graph classes, which extends the results of perfect EIC-graph of Cardoso et al. in [Injective edge chromatic index of a graph, http://arxiv.org/abs/1510.02626.].