Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs
type: Item
instance of: scholarly article
title: Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs
zbMATH Open document ID: 1481.05056
author: Fangfang Wu
author: Tingting Han
author: Shenggui Zhang
author: Binlong Li
published in: Discrete Applied Mathematics
publication date: 2021-12-07T00:00:00Z
review text: An edge-colored graph \(G^c\) consists of a graph \(G\) and a coloring mapping \(c\) which assigns colors to edges of \(G\). For a vertex \(v \in V(G^c)\), the color neighborhood of \(v\), denoted \(CN(v)\), is the set of colors used to color the neighbors of \(v\) in \(G\). The research investigates the sufficient conditions in terms of color neighborhood union to assure the existences of proper colored cycles with various lengths. The main results prove that for an edge-colored complete graph \(K_n^c\) on \(n\) vertices, for any integer \(k \ge 3\), if \(n\) is sufficiently large and if for any pair of distinct vertices \(u\) and \(v\), \(|CN(u) \cup CN(v)| \ge n\), then \(K_n^c\) has a properly colored cycle of length \(k\). Furthermore, if, in addition, \(K_n^c\) does not have a monochromatic triangle, then for any vertex \(v \in V(G^c)\) and for any integer \(k\) with \(3 \le k \le n\), there exists a properly colored cycle \(C\) of length \(k\) containing \(v\); and for any distinct vertices \(u,v\) and for any integer \(\ell\) with \(1 \le \ell \le n-1\), \(G^c\) contains a properly colored \((u,v)\)-path of length \(\ell\). A few open problems are raised in the concluding remark section of the paper, which seem quite interesting.
reviewed by: Hong-Jian Lai
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 68R10
zbMATH DE Number: 7439558
zbMATH Keywords: edge-colored complete graphs
zbMATH Keywords: color neighborhood
zbMATH Keywords: properly colored cycles
zbMATH Keywords: properly colored paths
MaRDI profile type: Publication
full work available at URL: j.dam.2021.10.016
OpenAlex ID: W3212087973
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DOI: 10.1016/J.DAM.2021.10.016
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description: scientific article; zbMATH DE number 7439558