Oriented vertex and arc colorings of partial 2-trees

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping φ from V (G) to V (H), that is φ(x)φ(y) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H. The oriented chromatic index of G is the minimum order of an oriented graph H such that the line-digraph of G has a homomorphism to H. In this paper, we determine the oriented chromatic number (resp. the oriented chromatic index) of the class of partial 2-trees for every girth g ≥ 3 (resp. for every girth g ≥ 3 excepted three). Introduction We consider finite simple oriented graphs, that is digraphs with no opposite arcs. For an oriented graph G, we denote by V (G) its set of vertices and by A(G) its set of arcs. The number of vertices of G is the order of G. The girth of a graph G is the size of a smallest cycle in G. We denote by Tg the class of partial 2-tree with girth at least g. The notion of oriented vertex-coloring was introduced by Courcelle [4] as follows: an oriented k-vertexcoloring of an oriented graph G is a mapping φ from V (G) to a set of k colors such that (i) φ(u) 6= φ(v) whenever uv ∈ A(G) and (ii) φ(v) 6= φ(x) whenever uv,xy ∈ A(G) and φ(u) = φ(y). The oriented chromatic number of G, denoted by χo(G), is defined as the smallest k such that G admits an oriented k-vertex-coloring. The oriented chromatic number χo(F) of a class of oriented graphs F is defined as the maximum of χo(G) taken over all graphs G in F. Let G and H be two oriented graphs. A homomorphism from G to H is a mapping φ from V (G) to V (H) that preserves the arcs: φ(u)φ(v) ∈ A(H) whenever uv ∈ A(G). An oriented k-vertex-coloring of an oriented graph G can be equivalently defined as a homomorphism φ from G to H, where H is an oriented graph of order k; such a homomorphism is called a H-vertex-coloring of G or simply a vertex-coloring of G. The existence of such a homomorphism from G to H is denoted by G→H. The vertices of H are called colors, and we say that G is H-vertex-colorable. The oriented chromatic number of G can then be equivalently defined as the smallest order of an oriented graph H such that G → H. Links between colorings and homomorphisms are presented in more details in the monograph [5] by Hell and Nešetřil. Oriented vertex-colorings have been studied by several authors in the last decade and the problem of bounding the oriented chromatic number has been investigated for various graph classes (see e.g. [3, 10, 11, 12]). Concerning partial 2-trees, Sopena proved [12] that their oriented chromatic number is at most 7 (this bound was shown to be tight). In [10], the authors obtained tight bounds for the oriented chromatic number of outerplanar graphs with given girth (which is a graph class strictly included in the class of partial 2-trees). Moreover, they proved that χo(Tg) = 7 for every g, 3 ≤ g ≤ 4. In this paper, we complete the characterization of the oriented chromatic numbers of partial 2-trees with given girth: Theorem 1 (1) χo(Tg) = 6 for every girth g, 5 ≤ g ≤ 6; (2) χo(Tg) = 5 for every girth g, g ≥ 7; ∗[email protected] †[email protected]