Rainbow Cycles in Properly Edge-Colored Graphs

We prove that every properly edge-colored n-vertex graph with average degree at least \(32(\log 5n)^2\) $32{(log5n)}^2$ contains a rainbow cycle, improving upon the \((\log n)^{2+o(1)}\) ${(logn)}^{2+o(1)}$ bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least \(10^5 k^3 n^{1+1/k}\) ${10}^5{k}^3{n}^{1+1/k}$ edges contains a rainbow 2k-cycle, which improves the previous bound \(2^{ck^2}n^{1+1/k}\) $2^{c{k}^2}{n}^{1+1/k}$ obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.