Proper Connection Number of Graph Products

Mao, Yaping; Yanling, Fengnan; Wang, Zhao; Ye, Chengfu

doi:10.1007/s40840-016-0442-z

Cited by 5 publications

(2 citation statements)

References 31 publications

(34 reference statements)

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“…The concept of pc(G) was first introduced by Borozan et al [4] and has been well-studied recently. We refer the reader to [2,4,7,14,19] for more details.…”

Section: Introductionmentioning

confidence: 99%

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Proper Vertex Connection and Graph Operations

Zhang

Zhu

2019

J. Inter. Net.

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A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by pvc k (G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertexconnected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. In this paper, we study the proper vertex k-connection number and the strong proper vertex-connection number on the join of two graphs, the Cartesian, lexicographic, strong and direct product, and present exact values or upper bounds for these operations of graphs.

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“…The concept of pc(G) was first introduced by Borozan et al [4] and has been well-studied recently. We refer the reader to [2,4,7,14,19] for more details.…”

Section: Introductionmentioning

confidence: 99%

“…where χ ′ (G) denotes the edge-chromatic number. Recently, the case for k = 1 has been studied by Andrews et al [2], Laforge et al [18] and Mao et al [25].…”

Section: Introductionmentioning

confidence: 99%