A basic randomized coloring procedure has been used in probabilistic proofs to obtain remarkably strong results on graph coloring. These results include the asymptotic version of the List Coloring Conjecture due to Kahn, the extensions of Brooks’ Theorem to sparse graphs due to Kim and Johansson, and Luby’s fast parallel and distributed algorithms for graph coloring. The most challenging aspect...

Many combinatorial problems can be efficiently solved for partial ktrees (graphs of treewidth bounded by k). The edge-coloring problem is one of the well-known combinatorial problems for which no NC algorithms have been obtained for partial k-trees. This paper gives an optimal and first NC parallel algorithm which finds an edge-coloring of a given partial k-tree using a minimum number of colors.

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...

Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ $k\ge 5$. The list problem asks following more general question: given available colors each vertex in graph, does it admit proper coloring? We propose quantum based Grover search to quadratically speed up exhaustive search...

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic number is determined for every complete multipartite graph as...

A graph G is k-colorable if G has a proper vertex coloring with k colors. The chromatic number (G) is the minimum number k such that G is k-colorable. A bcoloring of a graph with k colors is a proper coloring in which each color class contains a color dominating vertex. The largest positive integer k for which G has a b-coloring with k colors is the b-chromatic number of G, denoted by b(G). Th...

A graph G is called a complete k-partite (k ≥ 2) graph if its vertices can be partitioned into k independent sets V1, . . . , Vk such that each vertex in Vi is adjacent to all the other vertices in Vj for 1 ≤ i < j ≤ k. A complete k-partite graph G is a complete balanced kpartite graph if |V1| = |V2| = · · · = |Vk|. An edge-coloring of a graph G with colors 1, . . . , t is an interval t-colorin...

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex of G sees all k colors on its closed neighborhood. We denote Fall(G) the set of all positive integers k for which G has a fall k-coloring. In this paper, we study fall colorings of lexicographic product of graphs and categorical product of graphs and answer a question of [3] about fall colorings of categorical prod...

Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Co...

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 ...