Coloring directed cycles

Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0$, where $\lambda(\vec C)$ is the number of forward arcs minus the number of backward arcs in $\vec C$. This is not true. In this paper we show that $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0(\bmod~3)$ or $\vec C$ does not contain three consecutive arcs going in the same direction

Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

Nice labeling problem for event structures: a counterexample

In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ admits a labeling with at most $f(n)$ labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barth\'elemy and Constantin in 1993

A Study of kk-dipath Colourings of Oriented Graphs

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed $k$ and $t$, whether there exists such a $t$-colouring are proved.Comment: 14 page