Related papers: A Near-Optimal Kernel for a Coloring Problem
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…
In the \probrFix problem, we are given a graph $G$, a (non-proper) vertex-coloring $c : V(G) \to [r]$, and a positive integer $k$. The goal is to decide whether a proper $r$-coloring $c'$ is obtainable from $c$ by recoloring at most $k$…
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…
We investigate the List $H$-Coloring problem, the generalization of graph coloring that asks whether an input graph $G$ admits a homomorphism to the undirected graph $H$ (possibly with loops), such that each vertex $v \in V(G)$ is mapped to…
We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be…
A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed…
Many important science and engineering problems can be converted into NP-complete problems which are of significant importance in computer science and mathematics. Currently, neither existing classical nor quantum algorithms can solve these…
We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99],…
The list coloring problem is a variation of the classical vertex coloring problem, extensively studied in recent years, where each vertex has a restricted list of allowed colors, and having some variations as the $(\gamma,\mu)$-coloring,…
This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where…
We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our result…
Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively…
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal…
The maximum edge colouring problem considers the maximum colour assignment to edges of a graph under the condition that every vertex has at most a fixed number of distinct coloured edges incident on it. If that fixed number is $q$ we call…
We consider the problem of $q$-colouring a $k$-uniform random hypergraph, where $q,k \geq 3$, and determine the rigidity threshold. For edge densities above the rigidity threshold, we show that almost all solutions have a linear number of…
The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no…
A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex…
Let P_G(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing P_G(4) over all…
For fixed integers p and q, let f(n,p,q) denote the minimum number of colors needed to color all of the edges of the complete graph K_n such that no clique of p vertices spans fewer than q distinct colors. A construction is given which…