On the oriented chromatic number of dense graphs

Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $\delta$, and maximum degree $\Delta$. The \emph{oriented chromatic number} of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which $\delta\geq\log n$. We prove that every such graph has oriented chromatic number at least $\Omega(\sqrt{n})$. In the case that $\delta\geq(2+\epsilon)\log n$, this lower bound is improved to $\Omega(\sqrt{m})$. Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when $G$ is ($c\log n$)-regular for some constant $c>2$, in which case the oriented chromatic number is between $\Omega(\sqrt{n\log n})$ and $\mathcal{O}(\sqrt{n}\log n)$

Distance-two coloring of sparse graphs

Consider a graph $G = (V, E)$ and, for each vertex $v \in V$, a subset $\Sigma(v)$ of neighbors of $v$. A $\Sigma$-coloring is a coloring of the elements of $V$ so that vertices appearing together in some $\Sigma(v)$ receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set $\Sigma(v)$, denoted by $\rho(\Sigma)$. In this paper we study graph classes $F$ for which there is a function $f$, such that for any graph $G \in F$ and any $\Sigma$, there is a $\Sigma$-coloring using at most $f(\rho(\Sigma))$ colors. It is proved that if such a function exists for a class $F$, then $f$ can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes.Comment: 13 pages - revised versio