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The strong chromatic index of \(C_4\)-free graphs. (English) Zbl 0961.05022

The strong chromatic index of a graph \(G\) is the minimum number of induced matchings which partition \(E(G)\). In 1985, Erdős and Nešetřil conjectured that the strong chromatic index of every graph of maximum degree \(\Delta\) is at most \((5/4)\Delta^2\). Using a probabilistic method the author proves an asymptotically better result for graphs which do not have a cycle of length 4. Also, he shows that his bound is asymptotically best possible, up to a constant multiple. At the end, some open problems and a conjecture are stated.

MSC:

05C15 Coloring of graphs and hypergraphs
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