In 2017, Aharoni proposed the following generalization of the
Caccetta-H\"{a}ggkvist 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 βn/rβ.
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 rβ₯1,
there exists a constant crβ 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+crβ.Comment: 10 pages, 0 figures. Upgraded the main result from the previous
version so that it now holds up to an additive constan