6 problems

Let $G$ be a graph, and let $d$ be a nonnegative integer such that $G$ is $d$-degenerate, meaning that every subgraph of $G$ has a vertex of degree at most $d$. Let $\triangle$-fre…

Let $P$ be a $d$-regular edge-rooted pattern of depth $r$ and girth $2r+2$. For each edge-rooted pattern, let $\mathbf{e}(P)$ denote its constraint vector, and let…

Let $\rho(d,g)$ denote the maximum Hall ratio among graphs of maximum degree at most $d$ and girth at least $g$. Girth-shifting conjecture. The values presented in the table are up…

Let $G$ be a triangle-free graph with $m$ edges, and let $\chi_f(G)$ denote its fractional chromatic number. Edge-based fractional chromatic-number conjecture. As $m\to\infty$, eve…

Let $G$ be a triangle-free graph on $n$ vertices, and let $\chi_f(G)$ denote its fractional chromatic number. Fractional chromatic-number conjecture. As $n\to\infty$, every such gr…

Let $x\ge1$ be real and let $g\ge3$ be an integer. For a graph $G$, let $\chi_f(G)$ denote its fractional chromatic number, and let the girth of a graph be the length of its shorte…