Oriented coloring

Abstract: We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) defined as the minimum order of an oriented graph H such that G admits a homomorphism to H. We study the chromatic number of oriented k-trees and of oriented graphs with bounded degree. We show that there exist oriented k-trees with chromatic number at least 2k+1 - 1 and that every oriented k-tree has chromatic number at most (k + 1) × 2k. For 2-trees and 3-trees we decrease these upper bounds respectively to 7 and 16 and show that these new bounds are tight. As a particular case, we obtain that oriented outerplanar graphs have chromatic number at most 7 and that this bound is tight too. We then show that every oriented graph with maximum degree k has chromatic number at most (2k - 1) × 22k-2. For oriented graphs with maximum degree 2 we decrease this bound to 5 and show that this new bound is tight. For oriented graphs with maximum degree 3 we decrease this bound to 16 and conjecture that there exists no such connected graph with chromatic number greater than 7. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 191–205, 1997

Abstract: This paper is devoted to an oriented coloring problem motivated by a task assignment model. A recent result established the NP-completeness of deciding whether a digraph is k-oriented colorable; we extend this result to the classes of bipartite digraphs and circuit-free digraphs. Finally, we investigate the approximation of this problem: both positive and negative results are devised.

Abstract: A homomorphism from an oriented graph G to an oriented graph H is a mapping $\varphi$ from the set of vertices of G to the set of vertices of H such that $\buildrel {\longrightarrow}\over {\varphi (u) \varphi (v)}$ is an arc in H whenever $\buildrel {\longrightarrow}\over {uv}$ is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (the line digraph LD(G) of G is given by V(LD(G)) = A(G) and $\buildrel {\longrightarrow}\over {ab} \in A(LD(G))$ whenever $a=\buildrel {\longrightarrow}\over {uv}$ and $a=\buildrel {\longrightarrow}\over {vw}$). We give upper bounds for the oriented chromatic index of graphs with bounded acyclic chromatic number, of planar graphs and of graphs with bounded degree. We also consider lower and upper bounds of oriented chromatic number in terms of oriented chromatic index. We finally prove that the problem of deciding whether an oriented graph has oriented chromatic index at most k is polynomial time solvable if k ≤ 3 and is NP-complete if k ≥ 4. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 313–332, 2008

Abstract: A graph is planar if it can be embedded on the plane without edge-crossings. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239-249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215-219, 2005].