Related papers: Conditional Hardness for Approximate Coloring

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex…

An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that edge weighting…

An odd coloring of a graph $G$ is a proper vertex coloring $\varphi$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $\varphi^{-1}(c)\cap N(v)$ is odd. The concept of…

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper…

Let $G=(V, E)$ be a given edge-weighted graph and let its {\em realization} $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e \in E$ independently with probability $p$. In the {\em stochastic matching} problem, the goal…

The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs…

Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total $k$-coloring of $G$. Define a weight function on total coloring as $$\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$ where $N(x)=\{y\in V(G)|xy\in…

The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this note, we study the eternal game chromatic…

We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most beth_omega (nu)" proved by Kojman [6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing…

For a graph $G$ and $t,k\in\mathbb{Z}^+$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\rightarrow \binom{[k]}{t}$ such that $|f(v)\cap f(w)| < d(v,w)$ for all distinct $v,w \in V(G)$. The \emph{$t$-tone chromatic number} of…

An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph…

We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues…

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as…

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors.…

A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We…

Given a graph $G$, a colouring of $G$ is \emph{acyclic} if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $\chi_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of…

We study the following question: Given are two $k$-colorings $\alpha$ and $\beta$ of a graph $G$ on $n$ vertices, and integer $\ell$. The question is whether $\alpha$ can be modified into $\beta$, by recoloring vertices one at a time, while…

A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges…

In Defective Coloring we are given a graph $G = (V, E)$ and two integers $\chi_d, \Delta^*$ and are asked if we can partition $V$ into $\chi_d$ color classes, so that each class induces a graph of maximum degree $\Delta^*$. We investigate…

Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for…