On the oriented chromatic index of oriented graphs

Abstract

Abstract:A homomorphism from an oriented graph G to an oriented graph H is a mapping ϕ from the set of vertices of G to the set of vertices of H such thatis an arc in H whenever − → 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 − → ab ∈ A(LD(G)) whenever a = − → uv and b = − → 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 NPcomplete if k ≥ 4.

Chromatic Polynomials of Oriented Graphs

“…Many of the questions that have interested both applied and theoretical researchers in the study of graph colourings find an analogue in the study of oriented graphs. In addition to bounds for a variety of graph families [7,9,10,11,16], researchers have examined the computational complexity of related decision problems [2,13], the notion of clique for oriented graphs [3], oriented arc-colourings [15], f o (G, λ) = λ 7 − 6λ 6 + 10λ 5 + 6λ 4 − 34λ 3 + 33λ 2 − 10λ oriented list-colourings [23] and even an oriented colouring game [14]. An excellent overview of the state-of-the-art is given in [21].…”

Oriented vertex and arc colorings of outerplanar graphs

“…Ochem et al [8] proved the following: 9 , v 10 be an oriented 10-path. Any good T 4 -arc-coloring of P = P \ {v 2 , .…”

“…We consider two cases: We thus get a contradiction thanks to Proposition 6. We finally prove that for every k 3 there exist outerplanar graphs with girth k and oriented chromatic index at least 4, using a construction proposed in [8]. Observe first that any arc-coloring of a directed cycle of length p, p ≡ 1 or 2 (mod 3), must use at least 4 colors.…”

Oriented, 2-edge-colored, and 2-vertex-colored homomorphisms

“…The oriented chromatic index has been studied in [14,15,16]. The pushable chromatic number χ p (G o ) is the minimum of χ o (G o ) over the oriented graphs G o equivalent to G o under the push operation which reverses the arcs in an edge-cut of an oriented graph.…”