Decomposing edge-coloured graphs under colour degree constraints

Fujita, Shinya; Li, Ruonan; Wang, Guanghui

doi:10.1017/s0963548319000014

Cited by 5 publications

(1 citation statement)

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“…On one hand, oriented graphs are often used as auxiliary tools to find properly colored cycles. More details can be found in [7,13,14,17]. On the other hand, finding directed cycles can be formulated as a special case of finding properly colored cycles by the following construction from Li [13].…”

Section: Introductionmentioning

confidence: 99%

Properly colored short cycles in edge-colored graphs

Ding¹,

Hu²,

Wang³

et al. 2019

Preprint

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Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers s and t with t ≥ s ≥ 2, every edge-colored graph G with no properly colored K s,t contains a spanning subgraph H which admits an orientation D such that every directed cycle in D is a properly colored cycle in G. Using this result, we show that for r ≥ 4, if Caccetta-Häggkvist Conjecture holds , then every edge-colored graph of order n with minimum color degree at least n/r + 2 √ n + 1 contains a properly colored cycle of length at most r. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) K s,t .

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