Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic...

The paper is devoted to the model of compact cyclic edge-coloring. This variant of edge-coloring finds its applications in modeling schedules in production systems, in which production proceeds in a cyclic way. We point out optimal colorings for some graph classes and we construct graphs which cannot be colored in a compact cyclic manner. Moreover, we prove some theoretical properties of consid...

For planar graphs, we consider the problems of list edge coloring and list total coloring. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that ar...

We describe an approach to optimization based on a multiple-restart quasi-Hopfield network where the only problem-specific knowledge is embedded in the energy function that the algorithm tries to minimize. We apply this method to three different variants of the graph coloring problem: the minimum coloring problem, the spanning subgraph k-coloring problem, and the induced subgraph k-coloring pro...

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The c...

A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)}v∈V (G) of lists of colors to the...

For integers k, r > 0, a (k, r)-coloring of a graph G is a proper k-coloring c such that for any vertex v with degree d(v), v is adjacent to at least min{d(v), r} different colors. Such coloring is also called as an r-hued coloring. The r-hued chromatic number of G, χr (G), is the least integer k such that a (k, r)-coloring of G exists. In this paper, we proved that if G is a planar graph with ...

A good deal of research has been done and published on coloring of the vertices of graphs for several years while studying of the excellent work of those maestros, we get inspire to work on the vertex coloring of graphs in case of a particular triangular closed path structure what we achieve from the front view of a pyramidal structure. From here we achieve a repetitive nature of vertex colorin...

We prove three complexity results on vertex coloring problems restricted to Pk-free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P6-free graphs. Recent results of Hoàng et al. imply that this problem is polynomially solvable on P5-free graph...

A graph is H-free if it does not contain an induced subgraph isomorphic to H . We denote by Pk and Ck the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on kcoloring Pt-free graphs, and almost complete the classificatio...