6 problems

Let $G$ be a finite simple triangle-free graph, let $\chi(G)$ be its chromatic number, and let $\phi:V(G)\to[k]$ be a proper vertex coloring, where $k=\chi(G)$. A path in $G$ is ra…

Schrijver's conjecture. If $G$ is a properly edge-colored $d$-regular graph, then $G$ contains a rainbow path of length $d-1$.

Let $p$) be prime and let $a_1,a_2,\dots,a_d$ be distinct nonzero elements of $\mathbb{Z}_p$. A set of elements is rearrangeable if its elements can be ordered so that all partial…

Gallai-Ramsey reduction conjecture. For any graph $H$ with no isolated vertices, we have

Let $G$ be a connected $k$-chromatic graph. In a $k$-coloring, a certifying path is a path whose vertices meet the required distinct color classes, and call such a path forward or…

Let $G$ be a connected graph, let $\chi(G)$ denote its chromatic number, and call a path colorful under a proper $\chi(G)$-coloring when its $\chi(G)$ vertices have pairwise distin…