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 $\mathcal{O}(\sqrt{n}\log n)$

On the existence and non-existence of improper homomorphisms of oriented and 22-edge-coloured graphs to reflexive targets

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem

On Universal Graphs for Planar Oriented Graphs of a Given Girth

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . If each graph in a class K has a homomorphism to the same H 0 , then H 0 is K-universal. Let P k denote the class of orientations of planar graphs with girth at least k. Clearly, P 3 oe P 4 oe P 5 : : : We discuss the existence of P k -universal graphs with special properties. It is known (see [11]) that there exists a P 3 -universal graph on 80 vertices. We prove here that (1) there exist no planar P 4 -universal graphs; (2) there exists a planar P 16 -universal graph on 6 vertices; (3) for any k, there exist no planar P k -universal graphs of girth at least 6; (4) for any k, there exists a P 40k -universal graph of girth at least k + 1. This work was partially supported by the grants 97-01-01075 and 96-01-01614 of the Russian Foundation for Fundamental Research. y This work was partially supported by the grant 96-..