Subdivisions of a large clique in $C_6$-free graphs

Abstract

Mader conjectured that every $C_4$-free graph has a subdivision of a clique of order linear in its average degree. We show that every $C_6$-free graph has such a subdivision of a large clique. We also prove the dense case of Mader's conjecture in a stronger sense, i.e. for every $c$, there is a $c^{\prime }$ such that every $C_4$-free graph with average degree $c{n}^{1/2}$ has a subdivision of a clique $K_{\ell }$ with $\ell =\lfloor c^{\prime }{n}^{1/2}\rfloor$ where every edge is subdivided exactly $3$ times.

arXiv comment17 pages