3 problems

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…

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 $d4a2$ 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…