H -Free Coloring on Graphs with Bounded Tree-Width N. R. Aravind 1 , Subrahmanyam Kalyanasundaram 1 , and Anjeneya Swami Kare 2 1 Department of Computer Science and Engineering, IIT Hyderabad, Hyderabad, India {aravind,subruk}@iith.ac.in 2 School of Computer and Information Sciences, University of Hyderabad, Hyderabad, India askcs@uohyd.ac.in Abstract. Let H be a fixed undirected graph. A vertex coloring of an undirected input graph G is said to be an H-Free Coloring if none of the color classes contain H as an induced subgraph. The H-Free Chromatic Number of G is the minimum number of colors required for an H-Free Coloring of G. This problem is NP-complete and is expressible in monadic second order logic (MSOL). The MSOL formu- lation, together with Courcelle’s theorem implies linear time solvability on graphs with bounded tree-width. This approach yields an algorithm with running time f (||ϕ||,t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f (||ϕ||,t) on ||ϕ|| can be as bad as a tower of exponentials. In this paper, we provide an explicit combinatorial FPT algorithm to compute the H-Free Chromatic Number of a given graph G, param- eterized by the tree-width of G. The techniques are also used to provide an FPT algorithm when H is forbidden as a subgraph (not necessarily induced) in the color classes of G. 1 Introduction Let G be an undirected graph. The classical q-Coloring problem asks to color the vertices of the graph using at most q colors such that no pair of adjacent vertices are of the same color. The Chromatic Number of the graph is the minimum number of colors required for q-coloring the graph and is denoted by χ(G). The graph coloring problem has been extensively studied in various settings. In this paper we consider a generalization of the graph coloring problem called H-Free q-Coloring which asks to color the vertices of the graph using at most q colors such that none of the color classes contain H as an induced subgraph. Here, H is any fixed graph, |V (H)| = r, for some fixed r. The H-Free Chromatic Number is the minimum number of colors required to H-free color