The Erd\H{o}s--Faber--Lov\'{a}sz Conjecture revisited

TL;DR

This paper revisits the Erdős–Faber–Lovász Conjecture, providing an efficient coloring algorithm for a special case where shared vertices belong to exactly two cliques, and discusses related graph decomposition concepts.

Contribution

It introduces a simple coloring algorithm for a specific case of the conjecture and explores its connections with clique-decompositions and edge-colorings.

Findings

The conjecture holds when each shared vertex is in exactly two cliques.

An efficient algorithm is provided for this special case.

Connections to clique-decompositions and edge-colorings are discussed.

Abstract

The Erd\H{o}s--Faber--Lov\'{a}sz 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.