6 problems

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $K_{\chi(G)}$ be the complete graph on $\chi(G)$ vertices. A strong immersion of a graph $H$ in $G$ consists of…

Let $G$ be a finite simple graph, let $\alpha(G)$ denote its independence number, and let $\operatorname{had}(G)$ denote its Hadwiger number. Seymour's improvement conjecture. Ther…

Weak Hadwiger conjecture. Every graph $G$ with $\alpha(G)\leq 2$ contains

Füredi–Gyárfás–Simonyi conjecture.

Let $K_t$ denote the complete graph on $t$ vertices. A quasi-$K_t$-minor in a graph $G$ consists of $t$ nonempty, pairwise disjoint vertex sets $B_1,\ldots,B_t$ such that each unio…

Let $D$ be a graphic degree sequence, and let $\mathcal{R}(D)$ be the set of its simple graph realizations. Define $\chi(D)$ as the maximum chromatic number over graphs in…