相关论文: Maximum $k$- vs. $\ell$-colourings of graphs

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive…

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

The $k$-colouring reconfiguration problem asks whether, for a given graph $G$, two proper $k$-colourings $\alpha$ and $\beta$ of $G$, and a positive integer $\ell$, there exists a sequence of at most $\ell+1$ proper $k$-colourings of $G$…

In the $\ell$-Coloring Problem, we are given a graph on $n$ nodes, and tasked with determining if its vertices can be properly colored using $\ell$ colors. In this paper we study below-guarantee graph coloring, which tests whether an…

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n),…

A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is…

We present a polynomial-time algorithm that colors any 3-colorable $n$-vertex graph using $O(n^{0.19539})$ colors, improving upon the previous best bound of $\widetilde{O}(n^{0.19747})$ by Kawarabayashi, Thorup, and Yoneda [STOC 2024]. Our…

We obtain the following new coloring results: * A 3-colorable graph on $n$ vertices with maximum degree~$\Delta$ can be colored, in polynomial time, using $O((\Delta \log\Delta)^{1/3} \cdot\log{n})$ colors. This slightly improves an…

We consider the graph $k$-colouring problem encoded as a set of polynomial equations in the standard way over $0/1$-valued variables. We prove that there are bounded-degree graphs that do not have legal $k$-colourings but for which the…

Given an edge-colored graph, the goal of the proportional fair matching problem is to find a maximum weight matching while ensuring proportional representation (with respect to the number of edges) of each color. The colors may correspond…

We establish new algorithmic guarantees with matching hardness results for coloring and independent set problems in one-sided expanders and related classes of graphs. For example, given a $3$-colorable regular one-sided expander, we compute…

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no $(k-\epsilon)^{\operatorname{pw}(G)}\operatorname{poly}(n)$ time algorithm for deciding if an $n$-vertex graph $G$ with pathwidth $\operatorname{pw}(G)$ admits a proper vertex…

Given a graph $G$ that is modified by a sequence of edge insertions and deletions, we study the Maximum $k$-Edge Coloring problem Having access to $k$ colors, how can we color as many edges of $G$ as possible such that no two adjacent edges…

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor…

We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k > 3\Delta$, and returns a random proper $k$-coloring of $G$. The distribution of the coloring is…

We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $\Delta$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar…

For a graph $G$ define the parameters $\ell(G)$ and $L(G)$ as the minimum and maximum value of $\nu(G\backslash F)$, where $F$ is a maximum matching of $G$ and $\nu(G)$ is the matching number of $G$. In this paper, we show that there is a…

We study a generalization of the classical Hajnal-Szemer\'edi theorem to vertex-weighted graphs. Given a graph with nonnegative vertex weights, a coloring is called $\alpha$-approximately equitable up to one vertex ($\alpha$-EQ1) if, for…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at most $k$. We prove Brooks-type theorems for $k$-connected graphs with maximal local…