Proper vertex-pancyclicity of edge-colored complete graphs without monochromatic triangles

Chen, Xiaozheng; Huang, Fei; Yuan, Jinjiang

doi:10.1016/j.dam.2019.03.011

Cited by 14 publications

(6 citation statements)

References 20 publications

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“…Under the condition of Conjecture 1, Fujita and Magnant [7] proved that each vertex is contained in a PC triangle and a PC quadrangle; Li et al [10] showed that each vertex of G is contained in PC cycles of length at least δ c (G); Chen et al [6] confirmed the conjecture when no monochromatic triangles exist.…”

Section: Additional Results and Analysesmentioning

confidence: 97%

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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Section: Additional Results and Analysesmentioning

confidence: 97%

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

Li¹

2020

Preprint

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“…During the past few decades, the existence of cycles in graphs have been extensively studied in the literatures. We recommend Abouelaoualim et al (2010); Chen (2018); Chen andLi (2021, 2022); Chen et al (2019); Czygrinow et al (2021); Ehard and Mohr (2020); Fujita et al (2019); Guo et al (2022); Kano and Li (2008); for more results.…”

Section: Peixue Zhao Fei Huangmentioning

confidence: 99%

Rainbow vertex pair-pancyclicity of strongly edge-colored graphs

Peixue¹,

Huang²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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An edge-colored graph is \emph{rainbow }if no two edges of the graph have the same color. An edge-colored graph $G^c$ is called \emph{properly colored} if every two adjacent edges of $G^c$ receive distinct colors in $G^c$. A \emph{strongly edge-colored} graph is a proper edge-colored graph such that every path of length $3$ is rainbow. We call an edge-colored graph $G^c$ \emph{rainbow vertex pair-pancyclic} if any two vertices in $G^c$ are contained in a rainbow cycle of length $\ell$ for each $\ell$ with $3 \leq \ell \leq n$. In this paper, we show that every strongly edge-colored graph $G^c$ of order $n$ with minimum degree $\delta \geq \frac{2n}{3}+1$ is rainbow vertex pair-pancyclicity.

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“…In this context, it is worth mentioning that Abouelaoualim et al [1] established a sufficient condition in terms of the minimum monochromatic degree for the existence of compatible Hamilton cycles, when considering the existence of properly colored cycles of all possible lengths in edge-colored multigraphs. More results and problems (conjectures) related to the existence of compatible Hamilton cycles in specific edge-colored graphs can be found in [1,2,4,23,24,54,74,92].…”

Section: Subgraphs With Specific Coloring Patternsmentioning

confidence: 99%

Compatible spanning circuits in edge-colored graphs

Guo

et al. 2020

Discrete Mathematics

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Proper vertex-pancyclicity of edge-colored complete graphs without monochromatic triangles

Cited by 14 publications

References 20 publications

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

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

Rainbow vertex pair-pancyclicity of strongly edge-colored graphs

Compatible spanning circuits in edge-colored graphs

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