Rainbow clique subdivisions

TL;DR

The paper proves that sufficiently dense properly edge-colored graphs contain rainbow subdivisions of complete graphs, nearly matching the lower bounds, and extends results to rainbow Turán numbers of cycles.

Contribution

It establishes near-optimal conditions for the existence of rainbow clique subdivisions in properly edge-colored graphs, advancing understanding in rainbow graph theory.

Findings

Properly edge-colored graphs with high average degree contain rainbow clique subdivisions.

The bound on average degree is nearly tight, within a logarithmic factor.

Results imply bounds on rainbow Turán numbers for cycles.

Abstract

We show that for any integer $t \ge 2$, every properly edge colored $n$-vertex graph with average degree at least $(\log n)^{2+o(1)}$ contains a rainbow subdivision of a complete graph of size $t$. Note that this bound is within $(\log n)^{1+o(1)}$ factor of the lower bound. This also implies a result on the rainbow Tur\'{a}n number of cycles.