Related papers: A local epsilon version of Reed's Conjecture

Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several…

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $\Delta(G)\ge 9$ and its clique number satisfies $\omega(G)\le \Delta(G)-1$, then its chromatic number satisfies $\chi(G) \le \Delta(G)-1$. We prove this…

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced…

A graph $G=(V(G), E(G))$ is supereulerian if it has a spanning Eulerian subgraph. Let $\ell(G)$ be the maximum number of edges of spanning Eulerian subgraphs of a supereulerian graph $G$. In $1996$, Catlin conjectured that if $G$ is a…

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…

Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related…

The second author's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisties $\chi \leq \lceil \frac 12 (\Delta+1+\omega)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses…

A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the chromatic index…

An \emph{edge coloring} of a graph $G$ is strong if each color class is an induced matching of $G$. The \emph{strong chromatic index} of $G$, denoted by $\chi _{s}^{\prime }(G)$, is the minimum number of colors for which $G$ has a strong…

A subgraph $H$ of a multigraph $G$ is overfull if $ |E(H) | > \Delta(G) \lfloor |V(H)|/2 \rfloor$. Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. in 2012 formed the multigraph version of the…

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq…

Let $G$ be a claw-free graph on $n$ vertices with clique number $\omega$, and consider the chromatic number $\chi(G^2)$ of the square $G^2$ of $G$. Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the line graphs $L$ of simple…

Dvo\v{r}\'ak and Kawarabayashi [European Journal of Combinatorics, 2017] asked, what is the largest chromatic number attainable by a graph of treewidth $t$ with no $K_r$ subgraph? In this paper, we consider the fractional version of this…

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in…

In this note we consider a more general version of local sparsity introduced recently by Anderson, Kuchukova, and the author. In particular, we say a graph $G = (V, E)$ is $(k, r)$-locally-sparse if for each vertex $v \in V(G)$, the…

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…

In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the…

The biclique cover number $(\text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $\text{bc}(G) \geq \lceil \log_2(\text{mc}(G^c))…

The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $\Delta(G)=3$…

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega,…