Acyclic Coloring of Graphs

Coloring is an abstraction for partitioning a set of objects into a few independent sets. The notion of independence and the associated coloring rules vary from context to context. In the simplest case, adjacent vertices in a graph are required to receive different colors ( proper coloring). In parallel scientific computing, a proper coloring is used to identify computational subtasks that can be performed concurrently. A proper coloring of vertices of a graph G is acyclic if every 2-colored subgraph of G is acyclic. The acyclic coloring problem arises in the context of matrix partitioning for the estimation of the Hessian matrix associated with numerical optimization problems. The smallest number of colors required to acyclically vertex color a graph G is called its achromatic number and is denoted by a(G). Finding the achromatic number, a(G), of a graph(G) is an NP-hard problem. Several approaches have been used to solve this problem. Some researchers looked at finding the achroamtic number of family of graphs such as planar graphs, planar graphs with large girth, 1planar graphs, outerplanar graphs, and d-dimensional grids. Another direction of study is finding upper bounds on a(G) in terms of the maximum degree of G. Alon et al. used problabistic arguements and showed that it is possible to acyclically color any graph of maximum degree ∆, with O(∆ 4 3 ) colors. Fertin et al. designed a polynomial time algorithm to find upper bounds on a(G) in terms of ∆. This thesis is dedicated to designing deterministic algorithms to reduce upper bounds on a(G) for bounded degree graphs. By extending/generalizing ideas of Skulrattanakulchai we reduce upper bounds for a(G) for family of graphs of maximum degree five from 9 to 8 and, for the family of graphs of maximum degree 6 from 15 to 12 respectively. We show above result by extending a partial coloring by one vertex at a time. Let v be the vertex that we color during an iteration. During this process, in some scenarios it is required that we recolor some of the vertices already colored so as to make a color feasible for the vertex v, the vertex we try to color. However, note that this recoloring, if required, is limited to the neighborhood of the neighbors of v in all cases. We have presented a polynomial time algorithm to acyclically color the vertices of graphs with maximum degree ∆ using ∆ 2+2∆+2 3 colors. The algorithm improves the state-of-the-art by a factor of 3/2 to the best known result till date. Our algorithm is greedy, and at each time we obsevred up to 3-neighborhood of a vertex even though recoloring is done up to 2-neighborhood of vertex.