$L(j, \, k)$-Labelings of Kronecker Products of Complete Graphs

Haque, Md. Emdadul; Jha, Pranava K.

doi:10.1109/tcsii.2007.908918

Cited by 14 publications

(9 citation statements)

References 13 publications

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“…Proof If β 0 ≥ 3k, then by Lemma 1 (A), λ h,k (G 8 2k ≥ (h + 2k) + 2h + 4k = 3h + 6k. If α 1 < 2k + h, then there is v 2 ∈ A 1 such that f (v 2 ) = α 1 .…”

Section: Lemma 2 (A)mentioning

confidence: 94%

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L(h,k)-labelling for octagonal grid

Kim

Rho

Song

2014

International Journal of Computer Mathematics

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An L(h, k)-labelling of a graph G is an assignment of nonnegative integers, called labels, to the vertices of G such that two adjacent vertices receive labels that differ by at least h and those of distance two receive labels that differ by at least k. Among the span of all L(h, k)-labellings of G, λ h,k (G) is the smallest one. In this paper for the octagonal grid G 8 , we show that the upper bound for λ h,k (G 8 ) is 3h + 6k when h ≥ 4k. And when h > 6k, we show that λ h,k (G 8 ) = 3h + 6k.

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“…Proof If β 0 ≥ 3k, then by Lemma 1 (A), λ h,k (G 8 2k ≥ (h + 2k) + 2h + 4k = 3h + 6k. If α 1 < 2k + h, then there is v 2 ∈ A 1 such that f (v 2 ) = α 1 .…”

Section: Lemma 2 (A)mentioning

confidence: 94%

“…The lambda-number for the Cartesian product [10,16,17], the direct product [8][9][10][11] and the strong product [12] of some family of graphs have been studied.…”

Section: Introductionmentioning

confidence: 99%

L(h,k)-labelling for octagonal grid

Kim

Rho

Song

2014

International Journal of Computer Mathematics

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“…4 show that χ 2 (G 4,n ) = 6 if n = 4, 6. The horizontal (lateral) combination of two copies of P 4,4 and its horizontal combination with P 4,6 are 2-distance colorings with six colors of G 4,8 and G 4,14 , respectively and hence χ 2 (G 4,n ) = 6 if n = 8, 14.…”

Section: Propositionmentioning

confidence: 99%

“…The direct product G × H of two graphs G = (V G , E G ) and H = (V H , E H ) is the graph such that its vertex set is V G × V H and edge set is composed of {(u 1 , u 2 ), (v 1 , v 2 )} where {u 1 , v 1 } ∈ E G and {u 2 , v 2 } ∈ E H . The L(2, 1)-labeling problem for the Cartesian product [4,11,20,27,29] and the direct product of some families of graphs are studied by [8,11,10,12,17,21,23].…”

mentioning

confidence: 99%

2-distance colorings of some direct products of paths and cycles

Kim

Song

Rho

2015

Discrete Mathematics

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a b s t r a c tThe square of a graph is obtained by adding edges between vertices of distance two in the original graph. The 2-distance coloring problem of a graph is the vertex coloring problem of its square graph. Accordingly the chromatic number of 2-distance coloring is called the 2-distance chromatic number. The 2-distance coloring problem is equivalent to a kind of the distance two labeling problem, the L(1, 1)-labeling problem which is motivated by the channel assignment problem. In this paper we find the 2-distance chromatic number of the direct product of two cycles whose numbers of vertices are large enough. Moreover we find that also for the direct product of a path and a cycle.

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“…We conclude this section underlining that many technical papers have appeared, concerning the product of graphs. For example, in [45] the L(h, 1)-labelling of the Cartesian product of a cycle and a path is handled, while the authors of [93] determine the λ h,k -number of graphs that are the direct product of complete graphs, with certain conditions on h and k.…”

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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$L(j, \, k)$-Labelings of Kronecker Products of Complete Graphs

Cited by 14 publications

References 13 publications

L(h,k)-labelling for octagonal grid

L(h,k)-labelling for octagonal grid

2-distance colorings of some direct products of paths and cycles

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

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