On Aharoni’s rainbow generalization of the Caccetta–Häggkvist conjecture
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For a digraph G and v is an element of V(G), let delta(+)(v) be the number of out-neighbors of v in G. The Caccetta-Haggkvist conjecture states that for all k >= 1, if G is a digraph with n = |V(G)| such that delta(+)(v) >= k for all v is an element of V(G), then G contains a directed cycle of length at most [n/k]. In Aharoni et al. (2019), Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color classes, each of size k, has a rainbow cycle of length at most.n/k.. In this paper, we prove this conjecture if each color class has size Omega(k log k).
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Patrick Hompe, Petra Pelikánová, Aneta Pokorná, Sophie Spirkl (2021). On Aharoni’s rainbow generalization of the Caccetta–Häggkvist conjecture. UWSpace. http://hdl.handle.net/10012/18506
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