3 papers
math.CO2025
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…
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.GT2025
Crossing number inequalities for curves on surfaces
Alfredo Hubard, Hugo Parlier
We prove that, as grows, any family of homotopically distinct closed curves on a surface induces a number of crossings that grows at least like . We use this…