Nonuniform Degrees and Rainbow Versions of the Caccetta-Häggkvist Conjecture.

arxiv(2023)

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摘要
The Caccetta-Haggkvist conjecture (denoted CHC) states that the directed girth (the smallest length of a directed cycle) dgirth(D) of a directed graph D on n vertices is at most [n/delta+(D)], where delta+ (D) is the minimum outdegree of D. We consider a version involving all outdegrees, not merely the minimum one, and prove that if D does not contain a sink, then dgirth(D) <= 2 Sigma(upsilon is an element of V(D)) 1/deg+(upsilon)+1 In the spirit of a generalization of the CHC to rainbow cycles in [1], this suggests the conjecture that given nonempty sets F-1,., F-n, of edges of K-n, there exists a rainbow cycle of length at most 2 Sigma(1 <= i <= n) 1/|F-i|+1. We prove a bit stronger result when 1 <= |F-i| <= 2, thereby strengthening a result of DeVos et al. [J. Graph Theory, 96 (2021), pp. 192-202]. We prove a logarithmic bound on the rainbow girth in the case that the sets F-i are triangles.
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caccetta–häggkvist conjecture,rainbow versions,degrees
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