Related papers: Borel Vizing's Theorem for Graphs of Subexponentia…

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer…

We study the $(\Delta+1)$-edge-coloring problem in the parallel $\left(\mathrm{PRAM}\right)$ model of computation. The celebrated Vizing's theorem [Viz64] states that every simple graph $G = (V,E)$ can be properly $(\Delta+1)$-edge-colored.…

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red-blue edge coloring of an optimal 1-plane graph $G$ partitions the edge set of $G$…

We consider graph coloring and related problems in the distributed message-passing model. {Locally-iterative algorithms} are especially important in this setting. These are algorithms in which each vertex decides about its next color only…

As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $\Delta(H)$, then $H$ has a vertex $\Delta(H)$-coloring such that one of the color classes is a…

Let $G$ be a graph and $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is an edge coloring such that each color appears at each vertex $v\in V(G)$ at most $f (v)$ times. The minimum number of colors needed to…

We consider the problem of extending partial edge colorings of cartesian products of graphs. More specifically, we suggest the following Evans-type conjecture: If $G$ is a graph where every precoloring of at most $k$ precolored edges can be…

We consider a decentralized graph coloring model where each vertex only knows its own color and whether some neighbor has the same color as it. The networking community has studied this model extensively due to its applications to channel…

We show that, for every $k \ge 2$, every $k$-uniform hypergaph of degree $\Delta$ and girth at least $5$ is efficiently $(1+o(1) )(k-1) (\Delta / \ln \Delta )^{ 1/(k-1) } $-list colorable. As an application (and to the best of our…

Let $H$ be a triple system with maximum degree $d>1$ and let $r>10^7\sqrt{d}\log^{2}d$. Then $H$ has a proper vertex coloring with $r$ colors such that any two color classes differ in size by at most one. The bound on $r$ is sharp in order…

We show that for a number of parameterized problems for which only $2^{O(k)} n^{O(1)}$ time algorithms are known on general graphs, subexponential parameterized algorithms with running time $2^{O(k^{1-\frac{1}{1+\delta}} \log^2 k)}…

For a fixed graph $H$, the $H$-Coloring problem asks whether a given graph admits an edge-preserving function from its vertex set to that of $H$. A seminal theorem of Hell and Ne\v{s}et\v{r}il asserts that the $H$-Coloring problem is…

Graph coloring is a fundamental problem in computer science. We study the fully dynamic version of the problem in which the graph is undergoing edge insertions and deletions and we wish to maintain a vertex-coloring with small update time…

Brooks' Theorem [R. L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc.} 37:194-197, 1941] states that every graph $G$ with maximum degree $\Delta$, has a vertex-colouring with $\Delta$ colours, unless $G$ is a…

A {\em strong edge coloring} of a graph $G$ is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} $\chiup_{s}'(G)$ of a graph $G$ is the minimum number of colors in a strong edge…

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…

Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree $\Delta$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G is in at most $\Delta^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$ is…

In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects…

Several recent results from dynamic and sublinear graph coloring are surveyed. This problem is widely studied and has motivating applications like network topology control, constraint satisfaction, and real-time resource scheduling. Graph…

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich,…