The strong chromatic index of graphs and subdivisions

The strong chromatic index s′(G)s′(G) of a multigraph GG is the minimum integer kk such that there is an edge-coloring of GG with kk colors in which every color class is an induced matching. Let I(G)I(G) be a subdivision of a multigraph GG in which each edge of GG is subdivided exactly one time. Brualdi and Massey (1993) proved that s′(I(G))≤2Δ(G)s′(I(G))≤2Δ(G) for every simple graph GG. Let FDFD denote the graph obtained from a 5-cycle by adding D−2D−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 GG is not F3F3 and d(x)+d(y)≤5d(x)+d(y)≤5 for any edge xyxy of GG, then s′(G)≤6s′(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 GG has d(x)+d(y)≤D+2d(x)+d(y)≤D+2 with min{d(x),d(y)}≤2min{d(x),d(y)}≤2 for any edge xyxy of GG, and GG is not FDFD, then s′(G)≤2Ds′(G)≤2D. As a consequence, we have s′(I(G))≤2Δ(G)s′(I(G))≤2Δ(G) for every multigraph GG. Furthermore, s′(H)≤2Δ(G)s′(H)≤2Δ(G) for every subdivision HH of I(G)I(G) unless G=FΔ(G)G=FΔ(G).