Related papers: A note on Hadwiger's Conjecture for $W_5$-free gra…

We study a conjecture due to Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht stating that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor…

Let $G$ be a bipartite graph, and let $H$ be a bipartite graph with a fixed bipartition $(B_H,W_H)$. We consider three different, natural ways of forbidding $H$ as an induced subgraph in $G$. First, $G$ is $H$-free if it does not contain…

A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$…

Hadwiger conjectured in 1943 that for every integer $t \ge 1$, every graph with no $K_t$ minor is $(t-1)$-colorable. Kostochka, and independently Thomason, proved every graph with no $K_t$ minor is $O(t(\log t)^{1/2})$-colorable. Recently,…

A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing…

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the…

As evidence for the Odd Hadwiger Conjecture, Simonyi and Zsb\'an (2010) showed that every Kneser graph $G$ with large enough order (compared to $\chi(G)$) contains a totally odd subdivision of $K_{\chi(G)}$. A recent result of Steiner…

A graph $G$ is $k$-vertex-critical if $\chi(G) = k$ but $\chi(G-v)<k$ for all $v \in V(G)$. A graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A $W_4$ is the graph consisting of a $C_4$ plus an…

A graph $ G $ is called $ t $-tough if $ \left|S\right|\geq t\cdot w\left(G-S\right)$ for every cutset $ S $ of $G$. Chv\'atal conjectured that there exists a constant $ t_{0} $ such that every $ t_{0} $-tough graph has a hamiltonian cycle.…

The Conjecture of Hadwiger implies that the Hadwiger number $h$ times the independence number $\alpha$ of a graph is at least the number of vertices $n$ of the graph. In 1982 Duchet and Meyniel proved a weak version of the inequality,…

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…

Hadwiger's conjecture claims that any graph with no $K_t$ minor is $(t - 1)$-colorable. This has been proved for $t \le 6$, but remains open for $t \ge 7$. As a variant of this conjecture, graphs with no $K_t^=$ minor have been considered,…

Motivated by Hadwiger's conjecture, Seymour asked which graphs $H$ have the property that every non-null graph $G$ with no $H$ minor has a vertex of degree at most $|V(H)|-2$. We show that for every monotone graph family $\mathcal{F}$ with…

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…

Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and a graph $G$ we have $$\hom(H,G)\geq…

We prove that every graph $G$ for which $\omega(G) \geq 3/4(\Delta(G) + 1)$, has an independent set $I$ such that $\omega(G - I) < \omega(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $\omega(G) <…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has chromatic…

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, and all the paths…

The independence number $\alpha(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$…