Простые сборные графы: задачи, теоремы и приложения

A modern model of DNA information transfer (the epigenomic rearrangement model) describes the structure of rearranged DNA in terms of a 4-valent graph (all vertices, except possibly two, have valence 4), with a defined relation of neighbors for each edge at all vertices. This structure almost defines almost a dessin d'enfant: the surface in which the graph is embedded may be undirected. I will discuss some problems and conjectures related to such graphs and their applications, estimates of the genuses of the resulting dessins d'enfant, and our recent proof of the Angeleshka, Zhonoshka, and Saito conjecture, which describes the structure of graphs satisfying the condition of maximality for the number of Hamiltonian sets of polygonal paths.
This talk is based on the results of joint work with A. Guterman, N. Zhonoshka, A. Maksaev, and N. Ostroukhova.