A note on the Erdős-Farber-Lovász conjecture

A hypergraph H is linear if no two distinct edges of H intersect in more than one vertex and loopless if no edge has size one. A q-edge-colouring of H is a colouring of the edges of H with q colours such that intersecting edges receive different colours. We use ΔH to denote the maximum degree of H. A well-known conjecture of Erdős, Farber and Lovász is equivalent to the statement that every loopless linear hypergraph on n vertices can be n-edge-coloured. In this paper we show that the conjecture is true when the partial hypergraph S of H determined by the edges of size at least three can be ΔS-edge-coloured and satisfies ΔS⩽3. In particular, the conjecture holds when S is unimodular and ΔS⩽3.