In the (k, i)-coloring problem, we aim to assign sets of colors of size k to the vertices of a graph G, so that the sets which belong to adjacent vertices of G intersect in no more than i elements and the total number of colors used is minimum. This minimum number of colors is called the (k, i)-chromatic number. We present in this work a very simple linear time algorithm to compute an optimum (...

A vertex-distinguishing coloring of a graph G consists in an edge or a vertex coloring (not necessarily proper) of G leading to a labeling of the vertices of G, where all the vertices are distinguished by their labels. There are several possible rules for both the coloring and the labeling. For instance, in a set irregular edge coloring [5], the label of a vertex is the union of the colors of i...

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum Conflict-Free Coloring (Min-CF-Coloring). In its general form, the input of the Min-CF-coloring problem is a set system (X;S), where each S 2 S is a subset of X . The output is a coloring of the sets in S that satisfies the following constraint: for every x 2 X th...

Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. Recently, Ito et al. [5] proposed a framework of reconfiguration problems, and gave complexity and approximability results for reconfiguration problems derived from several well-known problems, such as independent set...

A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The `-Coloring problem is the problem to decide whether a graph can be colored with at most ` colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8free graphs. This improves a result of Le, Randerath, and Schierme...

This paper is concerned with the problem of supervised learning of deterministic finite state automata, in the technical sense of identification in the limit from complete data, by finding a minimal DFA consistent with the data (regular inference). We solve this problem by translating it in its entirety to a vertex coloring problem. Essentially, such a problem consists of two types of constrain...

In this paper we present a hybrid technique that applies a genetic algorithm followed by wisdom of artificial crowds approach to solving the graph-coloring problem. The genetic algorithm described here utilizes more than one parent selection and mutation methods depending on the state of fitness of its best solution. This results in shifting the solution to the global optimum more quickly than ...

We consider the interval constrained coloring problem, which appears in the interpretation of experimental data in biochemistry. Monitoring hydrogen-deuterium exchange rates via mass spectroscopy experiments is a method used in that field to obtain information about protein tertiary structure. The output of these experiments provides data about the exchange rate of residues in overlapping fragm...

Finding a proper coloring of a t-colorable graph G with t colors is a classic NP-hard problem when t ≥ 3. In this work, we investigate the approximate coloring problem in which the objective is to find a proper c-coloring of G where c ≥ t. We show that for all t ≥ 3, it is NP-hard to find a c-coloring when c ≤ 2t− 2. In the regime where t is small, this improves, via a unified approach, the pre...

Thomassen posed the following problem: Suppose G is a planar graph and W V (G) such that the distance between any two vertices in W is at least 100. Can a 5-coloring of W be extended to a 5-coloring of G? It is known that no 4-coloring result of this nature can hold. We provide a best possible solution to Thomassen's problem as well as some generalizations.