$(\Delta-1)$-dicolouring of digraphs

Abstract

In 1977, Borodin and Kostochka conjectured that every graph with maximum degree $\Delta \geq 9$ is $(\Delta-1)$-colourable, unless it contains a clique of size $\Delta$. In 1999, Reed confirmed the conjecture when $\Delta\geq 10^{14}$. We propose different generalisations of this conjecture for digraphs, and prove the analogue of Reed's result for each of them. The chromatic number and clique number are replaced respectively by the dichromatic number and the biclique number of digraphs. If $D$ is a digraph such that $\min(\tilde{\Delta}(D),\Delta^+(D)) = \Delta \geq 9$, we conjecture that $D$ has dichromatic number at most $\Delta-1$, unless either (i) $D$ contains a biclique of size $\Delta$, or (ii) $D$ contains a biclique $K$ of size $\Delta-2$, a directed $3$-cycle $\vec{C_3}$ disjoint from $K$, and all possible arcs in both directions between $\vec{C_3}$ and $K$. If true, this implies the conjecture of Borodin and Kostochka. We prove it when $\Delta$ is large enough, thereby generalising the result of Reed. We finally give a sufficient condition for a digraph $D$ to have dichromatic number at most $\Delta_{\min}(D)-1$, assuming that $\Delta_{\min}(D)$ is large enough. In particular, this holds when the underlying graph of $D$ has no clique of size $\Delta_{\min}(D)$, thus yielding a third independent generalisation of Reed's result. We further give a hardness result witnessing that our sufficient condition is best possible. To obtain these new upper bounds on the dichromatic number, we prove a dense decomposition lemma for digraphs having large maximum degree, which generalises to the directed setting the so-called dense decomposition of graphs due to Molloy and Reed. We believe this may be of independent interest, especially as a tool in various applications.