Structural parameterizations of budgeted graph coloring

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