The strong chromatic index of Halin graphs

A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index of a graph G , denoted by s χ ′ ( G ) , is the minimum number of colors needed for a strong edge coloring of G . A Halin graph G is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C . If a Halin graph G = T ∪ C is different from a certain necklace N e 2 and any wheel W n , n ≢ 0 ( mod 3 ) , then we prove that s χ ′ ( G ) ⩽ s χ ′ ( T ) + 3 .