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) ≤ 5 4 ∆ 2 for all graphs G with maximum degree ∆. The problem is far from being solved and the best known upper bound on s ′ (G) is 1.99∆ 2. We will discuss the topological variant of s ′ (G), denoted s ′ t (G) and called topological strong chromatic index. By a " topological relaxation " of the mentioned conjecture we mean the statement involving s ′ t (G) instead of s ′ (G). We show that for bipartite graphs G we have s ′ t (G) ≤ 1.703∆ 2. The proof uses the so-called K l,m-theorem and is purely combinatorial.