A classification of edge-colored graphs based on properly colored walks

Li, Ruonan; Li, Binlong; Zhang, Shenggui

doi:10.1016/j.dam.2020.02.008

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…It is easy to check that there is no PC cycle C with V (C) = V (C 1 ) ∪ V (C 2 ). Such a vertex set V (C 1 ) was called "degenerate set" by the first author when studying the relationship between edge-colored graphs and digraph (see [21,22]). Bánkfalvi and Bánkfalvi [4] gave a necessary and sufficient condition for the existence of a PC Hamilton cycle in a 2-colored K n .…”

Section: Discussionmentioning

confidence: 99%

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

Li¹,

Niu²

2022

Preprint

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A path (cycle) is properly-colored if consecutive edges are of distinct colors. In 1997, Bang-Jensen and Gutin conjectured a necessary and sufficient condition for the existence of a Hamilton path in an edge-colored complete graph. This conjecture, confirmed by Feng, Giesen, Guo, Gutin, Jensen and Rafley in 2006, was laterly playing an important role in Lo's asymptotical proof of Bollobás-Erdős' conjecture on properly-colored Hamilton cycles. In 1997, Yeo obtained a structural characterization of edge-colored graphs that containing no properly colored cycles. This result is a fundamental tool in the study of edge-colored graphs. In this paper, we first give a much shorter proof of the Bang-Jensen-Gutin Conjecture by two novel absorbing lemmas. We also prove a new sufficient condition for the existence of a properly-colored cycle and then deduce Yeo's theorem from this result and a closure concept in edge-colored graphs.

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Section: Discussionmentioning

confidence: 99%

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

Li¹,

Niu²

2022

Preprint

Self Cite

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“…Hence no PC Hamilton cycle exists. Generally, we give the following definition, which was firstly proposed in [11].…”

Section: Introductionmentioning

confidence: 99%

Properly colored cycles in edge-colored complete graphs containing no monochromatic triangles: a vertex-pancyclic analogous result

Li¹

2020

Preprint

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A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length 3 with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the exceptions.

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“…For convenience, we introduce a definition of degenerate sets given by Li et al [69]. Let G be an edge-colored complete graph, if there exists a nonempty set S ⊂ V (G) such that C(S) ⊆ C(S, V (G) \ S) and |C(S, V (G) \ S)| = 1, then we say that S is a 1-degenerate set of G; if there exists a nonempty set…”

Section: Introductionmentioning

confidence: 99%

Properly colored subgraphs in edge-colored graphs

Han

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A classification of edge-colored graphs based on properly colored walks

Cited by 5 publications

References 12 publications

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

Properly colored cycles in edge-colored complete graphs containing no monochromatic triangles: a vertex-pancyclic analogous result

Properly colored subgraphs in edge-colored graphs

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