5 problems

Let $S_g(n,m)$ be the random graph with $n$ vertices and $m$ edges embedded on the orientable surface $\mathbb{S}_g$, and let the giant component mean the component containing a li…

Let $S_g(n,m)$ be the random graph with $n$ vertices and $m$ edges embedded on an orientable surface of genus $g$, and let $H_1$ denote its largest component. Write … where…

Let $\Sigma$ be a surface of genus $g$, let $\mathcal{G}_\Sigma$ denote the class of graphs embeddable in $\Sigma$, and let $N(\Delta,k,\mathcal{G}_\Sigma)$ be the maximum order of…

Let $\Sigma$ be a surface, let $k\geq 2$ be an even diameter, and let $N(\Delta,k,\Sigma)$ denote the maximum order of a graph of maximum degree $\Delta$ and diameter $k$ embeddabl…

Chromatic-number conjecture. The chromatic number of a random graph of fixed genus $g$ with $n$ vertices is asymptotically almost surely equal to $4$.