An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented graph with maximum average degree strictly less than 10 3 has an oriented chromatic number at most 16. This implies that every oriented planar graph with girth at least 5 has an oriented chromatic number at most 16, that improves the previous known bound of 19 d...

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χa(G), is the least number of colors k such that G has an acyclic edge k-coloring. The maximum average degree of a graph G, denoted by mad(G), is the maximum of the average degree of all subgraphs of G. In this paper, it is proved that if mad(G) < 4, then χa(...

We analyze Markov chains for generating a random k-coloring of a random graph Gn,d/n. When the average degree d is constant, a random graph has maximum degree log n/ log log n, with high probability. We efficiently generate a random k-coloring when k = Ω(log log n/ log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but...

For distinct vertices u and v in a graph G, the connectivity between v, denoted $$\kappa _G(u,v)$$ , is maximum number of internally disjoint u–v paths G. The average $${\overline{\kappa }}(G),$$ taken over all unordered pairs u, Analogously, for directed D, from to _D(u,v)$$ D. }}(D)$$ ordered An orientation G obtained by assigning direction every edge let }}_{\max }(G)$$ denote among orientat...