Consecutive Colouring of Oriented Graphs

Abstract

We consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

A note on the interval colouring thickness of graphs

“…This allows us to deduce from Lemma 6 that θ(G) n α i+1 +o (1) . Clearly θ(n) n = n α 0 , so θ(n) n α i +o (1) and θ ′ (m) m β i +o (1) for all i ∈ N. The theorem now follows from the fact that α i → 5 6 and β i → 5 11 as i → ∞.…”

On interval colourings of graphs

“…Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol and Zuazua [7] defined an oriented graph to be consecutively colourable if there exists a proper arc colouring such that for each vertex v, the colours of all out-arcs from v and the colours of all in-arcs to v form two intervals of integers. Moreover they conjectured that for every graph G, there exists an orientation of the edges that is consecutively colourable.…”

“…More generally, every graph having an edge chromatic number greater than its maximum degree is not interval colourable. Interval colourings of some special classes of graphs have been investigated over the last years, as well as some related problems, see for instance [2,4,7,8,11,13,15].…”

Interval colorings of graphs—Coordinated and unstable no‐wait schedules

“…However, they are present on other two edges incident to v′. The respective vertices in V 0 are incident to two edges labeled by consecutive integers: D D (8, 7), (9,8), (10,9), (11,12), (12,11), …, ( + 19, + 20).…”

“…Indeed, otherwise considering the labels in an interval coloring modulo $${\rm{\Delta }}(G)$$ gives a proper edge‐coloring using at most $${\rm{\Delta }}(G)$$ colors. Interval colorings for special classes of graphs and related problems were considered, see, for example, [2–4, 6, 7, 9–13, 16–21, 24, 25].…”