On interval colouring reorientation number of oriented graphs

Abstract

We investigate a proper arc colouring of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals of integers. Oriented graphs having such colourings are called interval colourable. We analyse the parameter $\mathrm{icr}\left(D\right)$, which denotes the minimum number of arcs of $D$ that should be reversed so that a resulting oriented graph is interval colourable. We prove that for each non-negative integer $p$ there exists an oriented graph $D$ with the property $p\le \mathrm{icr}\left(D\right)\le 4p+1$. We show that $\mathrm{icr}\left(D\right)$ is not monotone with respect to taking subdigraphs. We give an upper bound on $\mathrm{icr}\left(D\right)$ if $D$ is an arbitrary oriented graph with finite parameter $\mathrm{icr}\left(D\right)$, and next if $D$ is any orientation of a 2-degenerate graph. Also the exact values of $\mathrm{icr}\left(D\right)$ for all orientations $D$ of some generalized Hertz graphs and generalized Sevastjanov rosettes are given. Based on special results concerning the still open problem of finding the largest possible transitive subtournament in a tournament (posed by Erdős and Moser in 1964) we refine the upper bound on $\mathrm{icr}\left(D\right)$ if $D$ is a tournament.