The oriented chromatic number of Halin graphs

Abstract

Oriented chromatic number of an oriented graph G is the minimum order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number of an unoriented graph G is the maximal chromatic number over all possible orientations of G. In this paper, we prove that every Halin graph has oriented chromatic number at most 8, improving a previous bound by Hosseini Dolama and Sopena, and confirming the conjecture given by Vignal.

Oriented colourings of graphs with maximum degree three and four

“…Upper bounds on the oriented chromatic number are known for orientations of graphs belonging to many graph families, for example partial t-trees [14], planar graphs [12], Halin graphs [5], outerplanar graphs [11], hypercubes [17], and grids [6].…”

Oriented cliques and colorings of graphs with low maximum degree

“…Hence, there is the arc 2 → 3 and so on, we show that there is the cycle 0 → 1 → 2 → ... → 12 → 0. Edges which are not oriented so far also form the cycle 0, 5,10,2,7,12,4,9,1,6,11,3,8. By Lemma 18a, this cycle must be oriented in one direction.…”

“…The notion of oriented coloring introduced by Courcelle [5] has been studied by several authors and the problem of bounding the oriented chromatic number has been investigated for various graph classes: outerplanar graphs (with given minimum girth) [18,20], 2-outerplanar graphs [9,16], planar graphs (with given minimum girth) [1,2,3,4,14,16,17,19], graphs with bounded maximum average degree [3,4], graphs with bounded degree [7,11,23], graphs with bounded treewidth [15,20,21], Halin graphs [8], graph subdivisions [25]. A survey on the study of oriented colorings has been done by Sopena in 2001 and recently updated [22].…”

Oriented chromatic number for Cartesian products Pm\Box Pn and Cm\Box Pn

“…Oriented coloring has been studied in recent years [1, 2, 6, 8-10, 12, 14, 16-20, 22], see [15] for a survey of the main results. Several authors established or bounded chromatic numbers for some families of graphs, such as oriented planar graphs [12,14], outerplanar graphs [12,17,18], graphs with bounded degree three [10,17,20], k-trees [17], Halin graphs [5,9], graphs with given excess [8] or grids [3,4,6,13,22].…”