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

A strong k -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 index χ s ′ ( G ) of a graph G is the minimum k for which G has a strong k -edge-coloring. Denote σ ( G ) = max x y ∈ E ( G ) { deg ( x ) + deg ( y ) − 1 } . It is easy to see that σ ( G ) ≤ χ s ′ ( 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 χ s ′ ( G ) ≤ 4 Δ + 4 by using the Four Color Theorem. The upper bound was then reduced to 4 Δ , 3 Δ + 5 , 3 Δ + 1 , 3 Δ , 2 Δ − 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. [ABSTRACT FROM AUTHOR]