A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping φ from V (G) to V (H), that is φ(x)φ(y) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H . The oriented chromatic index of G is the minimum order of an oriented graph H such that the line-digraph ...

A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, skew edge coloring of certain classes of graphs are determined. Furthermore, the skew chromatic index of those graphs is obtained ...

The weak chromatic number , or clique chromatic number (CCHN) of a graph is the minimum number of colors in a vertex coloring, such that every maximal clique gets at least two colors. The weak chromatic index , or clique chromatic index (CCHI) of a graph is the CCHN of its line graph. Most of the results here are upper bounds for the CCHI, as functions of some other graph parameters, and contra...

Given an oriented graph G, the modular flow polynomial φG(k) counts the number of nowhere-zero Zk-flows of G. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using Ehrhart–Macdonald reciprocity we give a combinatorial interpretation for the values of φG at negative arguments which answers a question of Beck and Zaslavsky (Adv Mat...

Abstract. A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, an algorithm is determined for skew edge coloring of circular ladder graphs. Alsothe skew chromatic index of circular la...

Vizing has shown that if G is a simple graph with maximum vertex-degree p, then the chromatic index of G is either p or p + 1 . In this note we prove that almost all graphs have a unique vertex of maximum degree, and we deduce that almost all graphs have chromatic index equal to their maximum degree . This settles a conjecture of the second author (in "Proceedings of the Fifth British Combinato...

Abstract The $\chi $ -stability index $\mathrm {es}_{\chi }(G)$ of a graph G is the minimum number its edges whose removal results in with chromatic smaller than that . We consider three open problems from Akbari et al. [‘Nordhaus–Gaddum and other bounds for edge-stability number’, European J. Combin. 84 (2020), Article no. 103042]. show by examples known characterisation k -regular ( $k\le 5$ ...