The Erd\H{o}s-Faber-Lovász Conjecture revisited


The Erd\H{o}s-Faber-Lovász Conjecture, posed in 1972, states that if a graph G is the union of n cliques of order n (referred to as defining n-cliques) such that two cliques can share at most one vertex, then the vertices of G can be properly coloured using n colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining n-cliques. We here provide a quick and easy algorithm to colour the vertices of G in this case, and discuss connections with clique-decompositions and edge-colourings of graphs.

DOI Code: 10.1285/i15900932v41n2p1

Keywords: Erdös-Faber-Lovász Conjecture; chromatic number; clique-decomposition; edge-colouring

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