Strong chromatic index in subset graphs (Q2713611)

The strong chromatic index \(\text{sq}(G)\) of a graph \(G\) is the minimum number of colors to color the edges of \(G\) such that each color class forms an induced matching in \(G\). For positive integers \(k\leq m\), the subset graph \(B_m(k)\) is the bipartite graph \((X,Y)\), where \(X\) is an \(m\)-element set and \(Y\) is the set of all \(k\)-element subsets of \(X\), and two vertices \(x\in X\), \(y\in Y\) are adjacent if and only if \(x\in y\). The main result of the paper proves that \(\text{sq}(B_m(k))= {m \choose k-1}\). This value satisfies a conjecture by \textit{R. A. Brualdi} and \textit{J. J. Q. Massey} [Discrete Math. 122, No. 1-3, 51-58 (1993; Zbl 0790.05026)].