Rainbow cycles for families of matchings
Ron Aharoni, He Guo
TL;DR
This work investigates rainbow cycles in edge-colored graphs, focusing on the rainbow girth when color classes are restricted to matchings rather than stars. The authors prove that if an $n$-vertex graph is colored with $n$ colors, each color class being a matching of size $2$, then the rainbow girth $rg(G)$ is $O(\log n)$, leveraging the Bollobás–Szemerédi sparse-graph girth bound and a probabilistic reduction to a small vertex subset that preserves a large rainbow edge set. The core technique combines a random subset $S$ selection with concentration inequalities (Chernoff and Chebyshev) to guarantee a subset of size $O(n)$ containing a rainbow edge set of size $\ge (c+\delta)n$, which in turn bounds the rainbow girth via the sparse-graph girth result. The paper further extends the bound to cases with fewer than $n$ color classes, establishing a log-scale bound for $\alpha n$ colors when $\alpha>3\sqrt{6}/8$ and identifying the critical threshold through a tangency argument, highlighting the structural difference between stars and matchings in color classes.
Abstract
Given a graph $G$ and a coloring of its edges, a subgraph of $G$ is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous Caccetta-Häggkvist conjecture, proposed by the first author, is that if in an coloring of the edge set of an $n$-vertex graph by $n$ colors, in which each color class is of size $k$, the rainbow girth is at most $\lceil \frac{n}{k} \rceil$. In the known examples for sharpness of this conjecture the color classes are stars, suggesting that when the color classes are matchings, the result may be improved. We show that the rainbow girth of $n$ matchings of size at least 2 is $O(\log n)$.