The acyclic chromatic number of a directed graph D, denoted χA(D), is the minimum positive integer k such that there exists a decomposition of the vertices of D into k disjoint sets, each of which induces an acyclic subgraph. We show that for all digraphs D without directed 2-cycles, we have χA(D) ≤ 4 5 · ∆̄(D) + o(∆̄(D)), where ∆̄(D) denotes the maximum arithmetic mean of the out-degree and the i...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a(G) ≤ ∆ + 2, where ∆ = ∆(G) denotes the maximum degree of the graph. A...

An (m,n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m,n)colored mixed graph G to an (m,n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is an arc (edge) of color c in H . The (m,n)-colored mixed chromatic number χ(m,n)(G) of...

An acyclic edge coloring of a graph G is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The acyclic chromatic index χa(G) of a graph G is the least number of colors needed in any acyclic edge coloring of G. Fiamčík (1978) conjectured that χa(G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G...

We provide a detailed study of topological and combinatorial properties sectionable tournaments. This class forms an inductively constructed family tournaments grounded over simply disconnected tournaments, those whose fundamental groups acyclic complexes are non-trivial. When T is tournament, we fully describe the cell-structure its complex Acy(T) by using adapted machinery discrete Morse theo...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contain...

A bound for the (n,m)-mixed chromatic number in terms of the chromatic number of the square of the underlying undirected graph is given. A similar bound holds when the chromatic number of the square is replaced by the injective chromatic number. When restricted to n = 1 andm = 0 (i.e., oriented graphs) this gives a new bound for the oriented chromatic number. In this case, a slightly improved b...

The maximum of the maximum degree and the 'odd set quotients' provides a well-known lower bound 4)(G) for the chromatic index of a multigraph G. Plantholt proved that if G is a multigraph of order at most 8, its chromatic index equals qS(G) and that if G is a multigraph of order 10, the chromatic index of G cannot exceed qS(G) + 1. We identify those multigraphs G of order 9 and 10 whose chromat...