Related papers: A Brooks' Theorem for Triangle-Free Graphs

Brooks' Theorem states that a connected graph $G$ of maximum degree $\Delta$ has chromatic number at most $\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson (1996) shows that if $G$ is triangle-free, then the…

We show that any triangle-free graph with maximum degree $\Delta$ has chromatic number at most $\left(1+o(1)\right)\Delta/\log \Delta.$

We prove that every triangle-free graph with maximum degree $\Delta$ has list chromatic number at most $(1+o(1))\frac{\Delta}{\ln \Delta}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that…

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in…

Problem of finding an optimal upper bound for the chromatic no. of a (3 Times K1)-free graph is still open and pretty hard. Here we prove that for a (3 Times K1)-free graph G with maximum degree greater than or equal to 8, {\chi} is less…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

A number of famous results about triangle-free graphs determine the maximum chromatic number of graphs of order n with minimum degree greater than n/3. In this note these results are extended to Kr-free graphs.

A triangle in a hypergraph $\mathcal{H}$ is a set of three distinct edges $e, f, g\in\mathcal{H}$ and three distinct vertices $u, v, w\in V(\mathcal{H})$ such that $\{u, v\}\subseteq e$, $\{v, w\}\subseteq f$, $\{w, u\}\subseteq g$ and…

Let $G$ be any triangle-free graph with maximum degree $\Delta\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the…

Circular $r$-coloring of a signed graph $(G,\sigma)$ is a mapping of its vertices to a circle of circumference $r$ such that: I. each pair of vertices with a negative connection is at distance at least $1$, and II. for each pair with a…

For a simple graph $G$, let $\chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $\chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $\Delta…

Here we prove that for a 2K2-free graph G with maximum degree greater than or equal to 5, the chromatic number is less than or equal to max{maximum degree-1, maximum clique size}. This implies that Borodin & Kostochka Conjecture is true for…

An $r$-hued coloring of a simple graph $G$ is a proper coloring of its vertices such that every vertex $v$ is adjacent to at least $\min\{r, \deg(v)\}$ differently colored vertices. The minimum number of colors needed for an $r$-hued…

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The…

We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G))…

King, Lu, and Peng recently proved that for $\Delta\geq 4$, any $K_\Delta$-free graph with maximum degree $\Delta$ has fractional chromatic number at most $\Delta-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$.…

By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $\Delta$ has chromatic number at most $(C+o(1))\Delta/\log \Delta$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one…

We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\Delta \geq 3$ can be lowered by 1, even in the case of online list coloring.

A graph $G$ is \emph{chordless} if no cycle in $G$ has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it…

Problem of finding an optimal upper bound for the chromatic no. of even 3K1-free graphs is still open and pretty hard. Here we prove Borodin & Kostochka Conjecture for 4K1-free graphs G i.e. If maximum degree of a {4 Times K1}-free graph is…