We present efficient branch and bound algorithms for solving the Maximum Clique Problem. First strategy is to store the size of clique consisting of a seed vertex in a subset induced by candidate set of the seed vertex. Results stored are further used to reduce the computation involved in computing clique consisting of other seed vertices in another subset of the graph. This approach along with...

In our work on robotic manipulation, we required a method for determining the orientation of a marked sphere from a single visual image. Our solution utilizes features of different colors painted on the sphere at the vertices of the Platonic solids. The main result of this paper is the minimization of the number of feature colors needed to solve the correspondence problem. The minimization of c...

Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between dominator chromatic number, chromat...

A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for each two colors i and j with i < j, every vertex of G colored by j has a neighbor with color i. The Grundy chromatic number Γ(G), is the largest integer k for which there exists a Grundy k-coloring for G. In this note we first give an interpretation of Γ(G) in terms of the total graph of G, when G is the complement of a...

The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree ∆ admits a (∆+2)-total-coloring. Similar to edge-colorings—with Vizing’s edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if ∆ ≥ 10, then every plane graph of maximum degree ∆ is (∆ + 1)...

Given an undirected graph G = (V, E) with a set V of vertices and a set E of edges, the graph coloring problem consists of partitioning all vertices into k independent sets and the number of used colors k is minimized. This paper presents a memetic algorithm (denoted by MACOL) for solving the problem of graph coloring. The proposed MACOL algorithm integrates several distinguished features such ...

We consider the bandwidth coloring problem, a generalization of the well-known graph coloring problem. For the latter problem, a classical theorem, discovered independently by Gallai, Roy and Vitaver, states that the chromatic number of a graph is bounded from above by the number of vertices in the longest elementary path in any directed graph derived by orienting all edges in the graph. We gen...

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prov...

This paper presents a Constraint Programming-based Column Generation approach to the Minimum Graph Coloring problem. The proposed approach has two versions of the pricing subproblems: the first version formulates the pricing subproblem as the optimization problem of finding the most negative reduced cost column, the second version as the decision problem of finding a maximal stable set with neg...

Graph/network theory results are applicable to problems in communications. As a representative example, the node coloring problem in graph theory is applicable to the channel assignment problem in cellular mobile communication systems. The node coloring problem is NP-complete, meaning that optimally solving it is very difficult. Therefore, we use heuristic algorithms for the channel assignment ...