The maximum size of an edge 2-neighborhood in P5-free graphs

Abstract

For a graph G=(V,E) and an edge uv∈E(G), the 2-neighborhood of uv is the set of all edges having at least one endvertex in N(u)∪N(v). A graph is called P5-free if it contains no induced subgraphs isomorphic to a path with 5 vertices. For P5-free graphs, we show that the maximum cardinality of an edge 2-neighborhood is at most 5Δ24, where Δ is the maximum degree of graphs. When Δ is even, this bound is tight and confirms the strong edge coloring conjecture of Erdős and Nešetřil for P5-free graphs.