On the oriented chromatic number of dense graphs

The oriented chromatic number of a graph G is the maximum, taken over all orientations of G, of the minimum number of colours in a proper vertex colouring of G 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, whose average degree is at least logarithmic in the number of vertices. For example, we prove that if G is (c log n)-regular for some constant c > 2, then the oriented chromatic number of G is between Ω( √ n log n) and O( √ n log n). Throughout this note, G is a (finite and simple) undirected graph with n vertices, m edges, and maximum degree ∆. A colouring of G is a function that assigns a ‘colour’ to each vertex so that adjacent vertices receive distinct colours. The chromatic number χ(G) is the minimum number of colours in a colouring of G. An orientation of G is a directed graph obtained from G by giving each edge one of the two possible orientations. A oriented colouring of D is a colouring of G such that between each pair of colour classes, all edges have the same direction. That is, there are no arcs −→ vw and −→ xy with c(v) = c(y) and c(w) = c(x). The oriented chromatic number −→χ (D) is the minimum number of colours in an oriented colouring of D. The oriented chromatic number −→χ (G) is the maximum of −→χ (D), taken over all orientations D of G. The oriented chromatic number is a widely studied parameter; see [1, 2, 3, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26]. This note is motivated by a question of André Raspaud [private communication, Prague 2004], who asked for the oriented chromatic number of the d-dimensional hypercube Qd. This is the graph with vertex set {0, 1}d, where two vertices are adjacent whenever they differ in precisely one coordinate. Qd is d-regular and has 2 d vertices. No non-trivial bounds on −→χ (Qd) were previously known. For example, Kostochka et al. [15] proved that −→χ (G) ≤ 2∆2 for all G, but this bound is more than the number of Date: March 26, 2008. 2000 Mathematics Subject Classification. 05C15 (coloring of graphs and hypergraphs).