On the precise value of the strong chromatic index of a planar graph with a large girth

A strongk-edge-coloring of a graph G is a mapping from E(G) to {1,2,…,k} such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic indexχs′(G) of a graph G is the minimum k for which G has a strong k-edge-coloring. Denote σ(G)=maxxy∈E(G){deg(x)+deg(y)−1}. It is easy to see that σ(G)≤χs′(G) for any graph G, and the equality holds when G is a tree. For a planar graph G of maximum degree Δ, it was proved that χs′(G)≤4Δ+4 by using the Four Color Theorem. The upper bound was then reduced to 4Δ, 3Δ+5, 3Δ+1, 3Δ, 2Δ−1 under different conditions for Δ and the girth. In this paper, we prove that if the girth of a planar graph G is large enough and σ(G)≥Δ(G)+2, then the strong chromatic index of G is precisely σ(G). This result reflects the intuition that a planar graph with a large girth locally looks like a tree.