Related papers: Coloring small locally sparse degenerate graphs an…
A hypergraph is "$d$-degenerate" if every subhypergraph has a vertex of degree at most $d$. A greedy algorithm colours every such hypergraph with at most $d+1$ colours. We show that this bound is tight, by constructing an $r$-uniform…
Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. On the other hand, if $G$ is weakly…
A well-known conjecture by Harris states that any triangle-free $d$-degenerate graph has fractional chromatic number at most $O\left(\frac{d}{\ln d}\right)$. This conjecture has gained much attention in recent years, and is known to have…
The weak degeneracy of a graph $G$ is a numerical parameter that was recently introduced by the first two authors with the aim of understanding the power of greedy algorithms for graph coloring. Every $d$-degenerate graph is weakly…
The defective chromatic number of a graph class is the infimum $k$ such that there exists an integer $d$ such that every graph in this class can be partitioned into at most $k$ induced subgraphs with maximum degree at most $d$. Finding the…
In recent work, Martinsson and Steiner showed that every $K_3$-free $d$-degenerate graph $G$ has fractional chromatic number $\chi_f(G) = O\left(\frac{d}{\log d}\right)$. In this paper, we extend the result in two ways, employing an…
Given a graph $G$ and a non-negative integer $d$ let $\alpha_d(G)$ be the order of a largest induced $d$-degenerate subgraph of $G$. We prove that for any pair of non-negative integers $k>d$, if $G$ is a $k$-degenerate graph, then…
A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete…
The classical Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem considers the chromatic number of $K_{r + 1}$-free graphs with large minimum degree, and in the case $r = 2$ says that any $n$-vertex triangle-free graph with minimum degree greater than…
A $b$-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the maximum…
We develop sequential algorithms for constructing edge-colorings of graphs and multigraphs efficiently and using few colors. Our primary focus is edge-coloring arbitrary simple graphs using $d+1$ colors, where $d$ is the largest vertex…
Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar…
\noindent The b-chromatic number of a graph $G$, denoted by $\phi(G)$, is the largest integer $k$ that $G$ admits a proper coloring by $k$ colors, such that each color class has a vertex that is adjacent to at least one vertex in each of…
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…
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed,…
We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $\Delta$ where each neighbourhood has density at most $d$, namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as…
We prove the following local strengthening of Shearer's classic bound on the independence number of triangle-free graphs: For every triangle-free graph $G$ there exists a probability distribution on its independent sets such that every…
Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ whenever $G$ is an $F$-free graph of maximum degree $\Delta$. The…
In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3,…
The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has…