The Erdős--Faber--Lovász Conjecture revisited

Abstract

The Erdős-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.