Parameterized Complexity of List Coloring and Max Coloring

Abstract

In the List Coloring problem, the input is a graph G and list of colors $L:V(G)\rightarrow {\mathbb N}$ for each vertex $v\in V(G)$. The objective is to test the existence of a coloring $\lambda :V(G)\rightarrow {\mathbb N}$ such that for each $v\in V(G)$, $\lambda (v)\in L(v)$ and for each edge $(u,v)\in E(G)$, $\lambda (u)\ne \lambda (v)$. Fiala et al. (TCS 2011) proved that List Coloring is W[1]-hard when parameterized by the vertex cover number of the input graph. Recently, Gutin et al. (STACS 2020, SIDMA 2021) designed an $O^*(2^k)$ time randomized algorithm for List Coloring where k is the size of the given clique modulator of the input graph. Since List Coloring is W[1]-hard parameterized by the vertex cover number, List Coloring is W[1]-hard parameterized by the size of a cluster modulator. In this work we study the problem parameteized by the size of $\ell$-cluster modulator. That is, along with the input we are also given a vertex subset D such that $G-D$ is cluster graph with $\ell$ connected components. We prove that assuming Exponential Time Hypothesis (ETH), List Coloring can not be solved in time $f(|D|+\ell )n^{o(\frac{|D|+\ell }{\log {|D|+\ell }})}$ for any computable function f.

In the Max Coloring problem, we are given a graph G and a weight function $w :V(G) \rightarrow {\mathbb N}$. For a proper coloring $\lambda$, the cost of $\lambda$ is defined as follows. For each color i, w(i) is the maximum weight of a vertex colored i. Then, the cost of $\lambda$ is $\sum _{i\in C} w(i)$, where $C=\{ \lambda (v) ~|~v\in V(G)\}$. In the Max Coloring problem, our objective is to find a proper coloring with minimum cost. Araújo et al. (TCS 2018) proved that Max Coloring is W[1]-hard even on forests when parameterized by the size of the largest connected component in the input forest. Here, we prove that Max Coloring is FPT with respect to parameters (i) the size of a vertex cover, (ii) the size of a clique modulator, and (iii) the size of a 2-cluster modulator.