8 problems

Aravind–Subramanian conjecture. There exists a constant $C$ such that

For a graph $G$ with $n$ vertices and genus $g$, let $\alpha_1(G)$ denote the order of a largest induced $1$-degenerate subgraph of $G$, equivalently a largest induced forest. Half…

For a graph $G$ and an orientable surface $S_t$ of genus $t$, let $\mu_t(G)$ be the skewness and let $\varepsilon_t(G):=\max\{0,\lceil\delta_t(G)\rceil\}$ be the integer Euler lowe…

Let $K_n$ be the complete graph, let $S_t$ be the orientable surface of genus $t$, and let $\delta_t(K_n)$ and $\mu_t(K_n)$ denote its Euler excess and skewness, respectively. Guy'…

Let $G$ be a complete graph or a complete bipartite graph, let $S_t$ be the orientable surface of genus $t$, and define its algebraic genus by … Here $\delta_t(G)$ is the Euler exc…

Let $\Sigma$ be a surface with finite genus and finitely many boundary components and punctures. A … -tight $\Sigma$-graph is irreducible if it has no digons, no triangles, and, fo…

Let $d4a2$ be a surface, and let $G$ be a graph embedded in $d4a2$. A list assignment $L$ assigns a set of colors to each vertex; $G$ is $L$-critical if it has no $L$-coloring but…

Let $S$ be a surface, and let $G$ be a graph that triangulates $S$. A graph satisfies Hajós' conjecture when, for every integer $k\geq 1$, every graph that is not $k$-colorable has…