On a topological relaxation of a conjecture of Erdős and Nešetřil

The strong chromatic index of a graph G, denoted by s′(G), is the minimum number of colors in a coloring of edges of G such that each color class is an induced matching. Erdős and Nešetřil conjectured that s′(G)≤54Δ2 for all graphs G with maximum degree Δ. The problem is far from being solved and the best known upper bound on s′(G) is 1.99Δ2, even in the case when G is bipartite.In this note we study the topological strong chromatic index, denoted by st′(G), defined as the Z2-index of a topological space obtained from the graph. It is known that s′(G)≥st′(G). We show that for bipartite graphs G we have st′(G)≤1.703Δ2.