Interval colourable orientations of graphs

Abstract

A graph is interval colourable if it admits a proper edge colouring in which for each vertex the set of colours of edges that are incident with the vertex is an interval of integers. An oriented graph is interval colourable if it admits a proper arc colouring in which for each vertex the set of colours of in-arcs and the set of colours of out-arcs that are incident with the vertex are both intervals of integers. In this paper we apply a classical approach and a new special trail technique to confirm the existence of interval colourable orientations of graphs from three classes: i) all k-trees with Δ(G)≤2k and k∈{2,3,4} and all k-paths with arbitrary k∈N; ii) graphs in which the length of every even closed trail is at least 2s and each such a trail has common edges with no more than d other even closed trails and 21−2se(d+1)≤1; iii) graphs that are decomposable into a bipartite interval colourable graph and a graph whose each connected component has at most one cycle that, if it exists, has odd length. Consequently, cactus graphs, graphs that are homeomorphic to Halin graphs, full subdivision graphs and others have interval colourable orientations.