Aharoni's rainbow cycle conjecture holds up to an additive constant

12/12/2022

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by   Patrick Hompe, et al.

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In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈ n/r ⌉. In this paper, we prove that, for fixed r, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed r ≥ 1, there exists a constant c_r such that if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most n/r + c_r.