5 problems

Let $G$ be a simple graph of order $n$, and let $F$ denote the number of faces of a uniformly random orientable embedding of $G$. Chain-of-triangles conjecture. For every such grap…

Let $G$ be an $n$-vertex multigraph with maximum edge-multiplicity $\mu$. Select an orientable embedding of $G$ uniformly at random, and let $F$ denote its number of faces. Multigr…

Let $G$ be a simple graph of order $n$. Select an orientable embedding of $G$ uniformly at random, and let $F$ denote its number of faces. Linear expected-face conjecture. The expe…

Let $G$ be a multigraph on $n$ vertices, and let $\mu$ be its maximum edge-multiplicity. Choose an orientable embedding of $G$ uniformly at random. Linear multigraph expected-face…

Let $G$ be a simple graph on $n$ vertices, and choose an orientable embedding of $G$ uniformly at random. Write $F_G$ for the number of faces of the resulting embedding. Linear exp…