Strong Chromatic Index in Subset Graphs

The strong chromatic index of a graph \(G\), denoted \(sq(G)\), is the minimum number of parts needed to partition the edges of \(G\) into induced matchings. The subset graph \(B_m(k)\) is the bipartite graph whose vertices represent the elements and the \(k\)-subsets of an \(m\) element ground set where two vertices are adjacent if and only if the vertices are distinct and the element corresponding to one vertex is contained in the subset corresponding to the other. We show that \(sq(B_m(k)) =\binom{m}{k-1}\) and that this satisfies the strong chromatic index conjecture by Brualdi and Quinn \([3]\) for bipartite graphs.