L(j, k)-number of direct product of path and cycle

Shiu, Wai Chee; Wu, Qiong

doi:10.1007/s10114-013-2021-7

Cited by 8 publications

(8 citation statements)

References 30 publications

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“…Note that, the distances between every pair of vertices in H and H are the same except that of v 4 and v 5 . Since the proof of Lemma 2.5 of Shiu and Wu (2013) did not involve the distance of v 4 and v 5 , we have λ j,k (H ) ≥ 3k + j. Apply the same labeling of H described in Shiu and Wu (2013) to H , we conclude that λ j,k (H ) = 3k + j.…”

Section: Lower Bounds On λ Jk (P N C M )mentioning

confidence: 83%

“…Proof By Lemma 2.5 in Shiu and Wu (2013), λ j,k (H ) = 3k + j. Note that, the distances between every pair of vertices in H and H are the same except that of v 4 and v 5 .…”

Section: Lower Bounds On λ Jk (P N C M )mentioning

confidence: 89%

“…L( j, k)-labeling numbers of graphs were studied in many articles, please refer to Calamoneri (2006); Jin (2007, 2005); Lam et al (2007); Liu (2001Liu ( , 2003; Zhu (2003, 2005); Mohar (2003); Yeh (2006); Zhu (2001) for j ≥ k and Bertossi and Bonuccelli (1995); Calamoneri et al (2006); Lin and Wu (2009) ;Jin and Yeh (2004); Niu (2007); Shiu and Wu (2013); Wu and Lin (2010); Wu et al (2014) for j ≤ k.…”

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confidence: 98%

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$$L(j,k)$$ L ( j , k ) -labeling number of Cartesian product of path and cycle

Shiu

Sun

2014

J Comb Optim

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For positive numbers j and k, an L( j, k)-labeling f of G is an assignment of numbers to vertices of G such thatThe span of f is the difference between the maximum and the minimum numbers assigned by f . The L( j, k)-labeling number of G, denoted by λ j,k (G), is the minimum span over all L( j, k)-labelings of G. In this article, we completely determine the L( j, k)-labeling number (2 j ≤ k) of the Cartesian product of path and cycle.

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Section: Lower Bounds On λ Jk (P N C M )mentioning

confidence: 83%

“…Proof By Lemma 2.5 in Shiu and Wu (2013), λ j,k (H ) = 3k + j. Note that, the distances between every pair of vertices in H and H are the same except that of v 4 and v 5 .…”

Section: Lower Bounds On λ Jk (P N C M )mentioning

confidence: 89%

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confidence: 98%

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$$L(j,k)$$ L ( j , k ) -labeling number of Cartesian product of path and cycle

Shiu

Sun

2014

J Comb Optim

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“…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%

“…Also the L(j, k)-number of the direct product of a path and a cycle for 2j ≤ k is obtained in [23]. In this paper, we find χ 2 (C m × C n ), the 2-distance chromatic number of C m × C n for sufficiently large m and n. In fact, we find 2-distance colorings of some C m × C n and then combine them to find a 2-distance coloring of C m × C n for sufficiently large m and n. Moreover we find χ 2 (P m × C n ), the 2-distance chromatic number of P m × C n by a similar method.…”

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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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Circular L(j,k)-Labeling Numbers of Cartesian Product of Three Paths

Rao

2022

Proceeding of 2021 International Conference on Wireless Communications, Networking and Applications

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The circular $$L(j,k)$$ L ( j , k ) -labeling problem with $$k\ge j$$ k ≥ j arose from the code assignment in the wireless network of computers. Given a graph $$G$$ G and positive numbers $$j,k,\sigma $$ j , k , σ , and a circular $$\sigma $$ σ -$$L(j,k)$$ L ( j , k ) -labeling of a graph $$G$$ G is an assignment $$f$$ f from $$[0,\sigma )$$ [ 0 , σ ) to the vertices of $$G$$ G , for any two vertices $$u$$ u and $$v$$ v , such that $$|f(u)-f(v){|}_{\sigma }\ge j$$ | f ( u ) - f ( v ) | σ ≥ j if $$uv\in E(G)$$ u v ∈ E ( G ) , and $$|f(u)-f(v){|}_{\sigma }\ge k$$ | f ( u ) - f ( v ) | σ ≥ k if $$u$$ u and $$v$$ v are distance two apart, where $$|f\left(u\right)-f\left(v\right){|}_{\sigma }=min\left\{|f(u)-f(v)|, \sigma -|f(u)-f(v)|\right\}$$ | f u - f v | σ = m i n | f ( u ) - f ( v ) | , σ - | f ( u ) - f ( v ) | . The minimum $$\sigma $$ σ such that graph $$G$$ G has a circular $$\sigma $$ σ -$$L(j,k)$$ L ( j , k ) -labeling of a graph $$G$$ G , which is called the circular $$L(j,k)$$ L ( j , k ) -labeling number of graph $$G$$ G and is denoted by $${\sigma }_{j,k}(G)$$ σ j , k ( G ) . In this paper, we determine the circular $$L(j,k)$$ L ( j , k ) -labeling numbers of Cartesian product of three paths, where $$k\ge 2j.$$ k ≥ 2 j .

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L(j, k)-number of direct product of path and cycle

Cited by 8 publications

References 30 publications

$$L(j,k)$$ L ( j , k ) -labeling number of Cartesian product of path and cycle

$$L(j,k)$$ L ( j , k ) -labeling number of Cartesian product of path and cycle

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

Circular L(j,k)-Labeling Numbers of Cartesian Product of Three Paths

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