Gleb Nenashev: On Heawood-type problem for maps with tangencies

Heawood type problems consist in estimating of the chromatic number of classes of graphs and maps.

Will talk about two problems: first for analog of $1$-planar on surfaces, i.e. graphs such that each edge intersects at most one other edge, and second for the class of maps on a surface of genus $g>0$ such that each point belongs to at most $k\geq 3$ regions. Second problem for $k=3$ is classical Heawood conjecture or Ringel–Youngs theorem.

Upper bounds on these chromatic numbers will presented in terms of genus. For case $k=4$ and for first problem, will be discussion about the lower bound.