An edge-coloured path is rainbow if its edges have distinct colours. An edge-coloured connected graph is said to be rainbow connected if any two vertices are connected by a rainbow path, and strongly rainbow connected if any two vertices are connected by a rainbow geodesic. The (strong) rainbow connection number of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the concept of rainbow vertexconnection. The topic of rainbow connection in graphs drew much attention and various similar parameters were introduced, mostly dealing with undirected graphs. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) rainbow vertex-connection number of digraphs. Results on the (strong) rainbow vertex-connection number of biorientations of graphs, cycle digraphs, circulant digraphs and tournaments are presented.

The star chromatic index of a multigraph G, denoted χ ′ s (G), is the minimum number of colors needed to properly color the edges of G such that no path or cycle of length four is bi-colored. A multigraph G is star k-edge-colorable if χ ′ s (G) ≤ k. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313-326] proved that every subcubic multigraph is star 7-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star 6-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether χ ′ s (G) ≤ 3 for an arbitrary graph G. This answers a question of Mohar. We then establish some structure results on subcubic multigraphs G with, where k ∈ {5, 6}. We finally apply the structure results, along with a simple discharging method, to prove that every subcubic multigraph G is star 6-edge-colorable if mad(G) < 5/2, and star 5-edge-colorable if mad(G) < 24/11, respectively, where mad(G) is the maximum average degree of a multigraph G. This partially confirms the conjecture of Dvořák, Mohar and Šámal.

An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) total rainbow connection number of digraphs. Results on the (strong) total rainbow connection number of biorientations of graphs, tournaments and cactus digraphs are presented.Comment: 29 pages, 4 figures. arXiv admin note: text overlap with arXiv:1701.0428

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