Structural parameterizations of budgeted graph coloring

We introduce a variant of the graph coloring problem, which we denote as BUDGETED COLORING PROBLEM (BCP). Given a graph G, an integer c and an ordered list of integers (b1, b2, ... , bc), BCP asks whether there exists a proper coloring of G where the i-th color is used to color at most bi many vertices. This problem generalizes two well-studied graph coloring problems, BOUNDED COLORING PROBLEM (BOCP) and EQUITABLE COLORING PROBLEM (ECP) and as in the case of other coloring problems, the BCP is NP-hard even for constant values of c. So we study BCP under the paradigm of parameterized complexity, particularly with respect to (structural) parameters that specify how far (the deletion distance) the input graph is from a tractable graph class.center dot We show that BCP is FPT (fixed-parameter tractable) parameterized by the vertex cover size. This generalizes a similar result for ECP and immediately extends to the BoCP, which was earlier not known.center dot We show that BCP is polynomial time solvable for cluster graphs generalizing a similar result for ECP. However, we show that BCP is FPT, but unlikely to have polynomial kernel, when parameterized by the deletion distance to clique, contrasting the linear kernel for ECP for the same parameter.center dot While the BoCP is known to be polynomial time solvable on split graphs, we show that BCP is NP-hard on split graphs. As BoCP is hard on bipartite graphs when c > 3, the result follows for BCP as well. We provide a dichotomy result by showing that BCP is polynomial time solvable on bipartite graphs when c = 2. We also show that BCP is NP-hard on co-cluster graphs, contrasting the polynomial time algorithm for ECP and BoCP.Finally we present an O*(2|V(G)|) algorithm for the BCP, generalizing the known algorithm with a similar bound for the standard chromatic number.(c) 2022 Elsevier B.V. All rights reserved.