The strong chromatic index of graphs and subdivisions

The strong chromatic index s ′ ( G ) of a multigraph G is the minimum integer k such that there is an edge-coloring of G with k colors in which every color class is an induced matching. Let I ( G ) be a subdivision of a multigraph G in which each edge of G is subdivided exactly one time. Brualdi and Massey (1993) proved that s ′ ( I ( G ) ) ≤ 2 Δ ( G ) for every simple graph G . Let F D denote the graph obtained from a 5-cycle by adding D − 2 new vertices and joining them to a pair of nonadjacent vertices of the 5-cycle. Wu and Lin (2008) proved that if a loopless multigraph G is not F 3 and d ( x ) + d ( y ) ≤ 5 for any edge x y of G , then s ′ ( G ) ≤ 6 . Their result solved the open problem proposed by Faudree et al. In this paper, we show a stronger form of both aforementioned results. We prove that if a loopless multigraph G has d ( x ) + d ( y ) ≤ D + 2 with min { d ( x ) , d ( y ) } ≤ 2 for any edge x y of G , and G is not F D , then s ′ ( G ) ≤ 2 D . As a consequence, we have s ′ ( I ( G ) ) ≤ 2 Δ ( G ) for every multigraph G . Furthermore, s ′ ( H ) ≤ 2 Δ ( G ) for every subdivision H of I ( G ) unless G = F Δ ( G ) .