1-planar graphs with minimum degree at least 3 have bounded girth

Abstract

We show that every 1-planar graph with minimum degree at least 4 has girth at most $8$, and every 1-planar graph with minimum degree at least 3 has girth at most $198$.

It has come to the attention of the author that the main result of the contribution can readily be obtained by combining existing results. Graphs with minimum degree 3 and large enough girth have a 2d-shallow minor of minimum degree at least $2^d$. Moreover, the edge density in shallow minors of $k$-planar graphs is polynomial for any fixed $k$ (and thus for $k=1$)