Improved Bounds on the $L(2,1)$-Number of Direct and Strong Products of Graphs

Shao, Zhang; Klavžar, Sandi; Shiu, Wai Chee; Zhang, David

doi:10.1109/tcsii.2008.921411

Cited by 11 publications

(6 citation statements)

References 22 publications

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“…The authors prove that for all the three classes of graphs the conjecture by Griggs and Yeh is true. In [164] the previous upper bounds for direct and strong product of graphs are improved. The main tool for this purpose is a more refined analysis of neighborhoods in product graphs.…”

Section: Product Of Paths Cycles and Cliquesmentioning

confidence: 99%

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Calamoneri

2011

The Computer Journal

160

104

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Given any fixed nonnegative integer values h and k, the L(h, k)labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with minimum span. The L(h, k)-labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.

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Section: Product Of Paths Cycles and Cliquesmentioning

confidence: 99%

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Calamoneri

2011

The Computer Journal

160

104

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“…The (2,1) L -labeling has been extensively studied in recent past by many researchers [see, 1,2,4,7,8,[14][15][16][17][18][19][20][21]. The common trend in most of the research paper is either to determine the value of (2,1) L -labeling number or to suggest bounds for particular classes of graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Klavzar and Spacepan [9] have shown that 2 ∆ -conjecture holds for graphs that are direct or strong products of nontrivial graphs. After that Shao, et al [15] have improved bounds on the (2,1) L -labeling number of direct and strong product of nontrivial graphs with refined approaches. Shao and Zhang [17] also consider the graph formed by the Cartesian sum of graphs and prove that the λ -number of (2,1) L -labeling of this graph satisfies the 2 ∆ -conjecture (with minor exceptions).…”

Section: Introductionmentioning

confidence: 99%

The L(2, 1)-labeling on <i>β</i>-product of Graphs

Kumar¹

2018

IJDST

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The L(2, 1)-labeling (or distance two labeling) of a graph G is an integer labeling of G in which two vertices at distance one from each other must have labels differing by at least 2 and those vertices at distance two must differ by at least 1. The L(2, 1)-labeling number of G is the smallest number k such that G has an L(2, 1)-labeling with maximum of f(v) is equal to k, where v belongs to vertex set of G. In this paper, upper bound for the L(2, 1)-labeling number for the β-product of two graphs has been obtained in terms of the maximum degrees of the graphs involved.

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“…The L(2, 1)-labelling and the circular-L(2, 1)-labelling on products of graphs have been rather extensively studied in recent years [6,8,12,15,17,18].…”

Section: Introductionmentioning

confidence: 99%