Let G = (V, E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1, . . . , k}) to each vertex of V so that no edge has both endpoints with the same color. We describe in this paper a new ant algorithm for the k-coloring problem. Computational experiments give evidence that our algorithm is competitive with the existing ant algorithms ...

An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex x are colored by d_{G}(x) consecutive colors, where d_{G}(x) is the degree of the vertex x in G. For Mobius ladders the existence of this coloring is proved and all possible numbers of colors in su...

In this paper we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. This is the second paper in a series of two. The first one, [3] is also submitted to this journal. In [2, 3], a polynomial time algorithm is given for three-coloring triangle-free graphs with no i...

We show two results related to the Hamiltonicity and k-Path algorithms in undirected graphs by Björklund [FOCS’10], and Björklund et al., [arXiv’10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O(1.6572) time. It can be applied as a subroutine to solve the k-Internal Spanning Tree (k-IST) problem...

Graph vertex coloring is one of the most studied NP-hard combinatorial optimization problems. Given the hardness of the problem, various heuristic algorithms have been proposed for practical graph coloring, based on local search, population-based approaches and hybrid methods. The research in graph coloring heuristics is very active and improved results have been obtained recently, notably for ...

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

For a fixed simple digraph F and given D , an - free k -coloring of is vertex-coloring in which no induced copy monochromatic. We study the complexity deciding for whether admits -free -coloring. Our main focus on restriction problem to planar input digraphs, where it only interesting cases ? { 2 3 } . From known results follows that every whose underlying graph not forest, 2-coloring, with ? (...

The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some proper...

The harmonious coloring of an undirected simple graph is a vertex coloring such that adjacent vertices are assigned different colors and each pair of colors appears together on at most one edge. The harmonious chromatic number of a graph is the least number of colors used in such a coloring. The harmonious chromatic number of a path is known, whereas the problem to find the harmonious chromatic...

An interval edge t − coloring of a graph G is a proper edge coloring of G with colors 1, 2, , t ... such that at least one edge of G is colored by color , 1, 2, , i i t = ... , and the edges incident with each vertex ( ) v V G ∈ are colored by ( ) G d v consecutive colors, where ( ) G d v is the degree of the vertex v in G . In this paper interval edge colorings of bipartite cylinders and bipar...