Related papers: Towards the Overfull Conjecture

Let $G=(V,E)$ be a multigraph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\frac{3}{2}\Delta$ colors by Shannon's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$…

A collection of graphs is \textit{nearly disjoint} if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every…

Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most…

In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the…

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a simple connected graph $G$ is either $\Delta(G)+1$ or $\Delta(G)+2$, where $\Delta(G)$ is the largest degree of any vertex…

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…

For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\psi:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrow\psi(i)\psi(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all…

The problem of chromatic thresholds seeks for minimum degree conditions that ensure $H$-free graphs to have a bounded chromatic number, or equivalently a bounded size homomorphic image. The strengthened homomorphism thresholds problem…

Let $G$ be a graph of order $n$, maximum degree at most $\Delta$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function…

For a hypergraph $G$, let $\chi(G), \Delta(G),$ and $\lambda(G)$ denote the chromatic number, the maximum degree, and the maximum local edge connectivity of $G$, respectively. A result of Rhys Price Jones from 1975 says that every connected…

Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of $G$ satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known…

Given two graphs $H$ and $G$, an $H$-tiling is a family of vertex-disjoint copies of $H$ in $G$. A perfect $H$-tiling covers all vertices of $G$. The Corradi-Hajnal theorem (1963) states that an $n$-vertex graph $G$ with minimum degree…

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend…

An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors…

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of…

In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($\Delta$ + 1)-edge-coloring of G where $\Delta$ is the maximum degree of G. One year later he conjectured that one can also…

Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection of vertex-disjoint copies of $H$ spanning $G$. K\"uhn and Osthus showed that the minimum degree threshold for a graph $G$ to contain a perfect $H$-factor is either given…

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and…

We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree…

Borodin and Kostochka conjectured that every graph $G$ with maximum degree $\Delta \ge 9$ satisfies $\chi \le \max\{\omega, \Delta-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main…