For positive integers k and r , a (k, r)-coloring of a graphG is a proper coloring of the vertices with k colors such that every vertex of degree i will be adjacent to vertices with at least min{i, r} different colors. The r-dynamic chromatic number of G, denoted by χr (G), is the smallest integer k for which G has a (k, r)-coloring. For a k-list assignment L to vertices of G, an (L, r)-colorin...

In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by C...

In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by C...

The Square Coloring of a graph G refers to coloring vertices such that any two distinct which are at distance most receive different colors. In this paper, we initiate the study related problem called subset square graphs. Broadly, studies dominating set using q Here aim is optimize number colors used. This also generalizes well-studied Efficient Dominating Set problem. We show $$q$$ -Subset NP...

We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Qd, and lists of allowed colors for the non-colored edges of Qd, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy ...

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...