An extension of the channel-assignment problem: L(2, 1)-labelings of generalized Petersen graphs

Adams, Sarah Spence; Cass, Jonathan; Troxell, Denise Sakai

doi:10.1109/tcsi.2005.862184

Cited by 14 publications

(8 citation statements)

References 18 publications

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“…Hence, in such cases a good aproximation for the bound of Theorem 3.1 is given by . In [14] it is proved that (1) We conclude the section by demonstrating that the upper bound of Theorem 3.1 is an improvement of (1). Because , we have thus reduced (1) by .…”

Section: Direct Product Of Graphsmentioning

confidence: 71%

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

Shao

Klavžar

Shiu

et al. 2008

IEEE Trans. Circuits Syst. II

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Abstract-The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An (2 1)-labeling of a graph is a function from the vertex set ( ) to the set of all nonnegative integers such that ( ) . This paper considers the graph formed by the direct product and the strong product of two graphs and gets better bounds than those of Klavžar andŠpacapan with refined approaches.

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Section: Direct Product Of Graphsmentioning

confidence: 71%

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

Shao

Klavžar

Shiu

et al. 2008

IEEE Trans. Circuits Syst. II

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“…This is not the case, as proved in [126], where it is shown that for every h ≥ 1 there exist ∆ 0 such that every K 4 -minor free graph with maximum degree ∆ ≥ ∆ 0 has an L(h, 1)-labelling with span at most ⌊ 3 2 ∆⌋ and this bound cannot be further decreased. This result translates to L(h, k)-labelling, with k > 1 providing an upper bound for λ h,k of k⌊ 3 2 ∆⌋.…”

Section: Open Problem: It Remains An Open Problemmentioning

confidence: 92%

“…7 is an upper bound also for generalized Petersen graphs of order greater than 6. In [3,4] the authors prove that this conjecture is true for orders 7 and 8, and give exact λ 2,1 -numbers for all generalized Petersen graphs of orders 5, 7 and 8, thereby closing all cases with orders up to 8. Finally, in [5] the exact λ 2,1 -numbers for all generalized Petersen graphs of orders 9, 10, 11 and 12 are given, thereby closing all open cases up to order N = 12 and lowering the upper bound on λ 2,1 down to 6 for all but three graphs of these orders.…”

Section: Generalized Petersen Graphsmentioning

confidence: 95%

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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“…After their work, λ 2 1 (G) is called the λ-number of G and denoted by λ(G). The λ-number of some classes of graphs have been studied in many literatures [1,13,12,4,10]. See [2,3,5] for extensive surveys.…”

Section: Introductionmentioning

confidence: 99%

The Λ-Number OF THE CARTESIAN PRODUCT OF a COMPLETE GRAPH AND a CYCLE

Kim¹,

Song²,

Rho³

2013

Korean Journal of Mathematics

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An L(j, k)-labeling of a graph G is a vertex labeling such that the difference of the labels of any adjacent vertices is at least j and that of any vertices of distance two is at least k for given j and k. The minimum span of all L(2, 1)-labelings of G is called the λ-number of G and is denoted by λ(G). In this paper, we find a lower bound of the λ-number of the Cartesian product K m C n of the complete graph K m of order m and the cycle C n of order n. In fact, we show that when n ≥ 3, λ(K 4 C n) ≥ 7 and the equality holds if and only if n is a multiple of 8. Moreover when m ≥ 5, λ(K m C n) ≥ 2m − 1 and the equality holds if and only if n is even.

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An extension of the channel-assignment problem: L(2, 1)-labelings of generalized Petersen graphs

Cited by 14 publications

References 18 publications

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

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

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

The Λ-Number OF THE CARTESIAN PRODUCT OF a COMPLETE GRAPH AND a CYCLE

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