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math.CO2025

Crossing numbers of dense graphs on surfaces

Alfredo Hubard, Arnaud de Mesmay, Hugo Parlier

In this paper, we provide upper and lower bounds on the crossing numbers of dense graphs on surfaces, which match up to constant factors. First, we prove that if is a dense eno…

math.CO2024

Crossing lemmas for -systems of arcs

Alfredo Hubard, Hugo Parlier

We show a generalization of the crossing lemma for multi-graphs drawn on orientable surfaces in which pairs of edges are assumed to be drawn by non-homotopic simple arcs which pair…