The channel assignment problem for mutually adjacent sites

Griggs, Jerrold R.; Liu, Daphne Der-Fen

doi:10.1016/0097-3165(94)90096-5

Cited by 30 publications

(23 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The parameter rtðTÞ has been studied by Rabinowitz and Proulx [22] and by Griggs and Liu [14]. It was noted in Ref.…”

Section: Almost Difference Closed Sets Of Type A2mentioning

confidence: 98%

“…It was noted in Ref. [14] that ðMÞ ¼ rtðTÞ, if T ¼ M [ f0g. Some known results used in this section are originally given in terms of rtðTÞ.…”

Section: Almost Difference Closed Sets Of Type A2mentioning

confidence: 99%

“…In 1973, Cantor and Gordon [2] proved the existence of ðMÞ for any M. In addition to Ref. [2], the values of ðMÞ for several special families of M have also been studied by Haralambis [16], Rabinowitz and Proulx [22], and Griggs and Liu [14].…”

Section: Introductionmentioning

confidence: 96%

“…By some result in Ref. [14] and [2], we may assume that S is periodic, i.e., there is an integer n such that for any k ! 0, S \ fkn; kn þ 1; .…”

mentioning

confidence: 99%

See 3 more Smart Citations

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

Liu

Zhu

2004

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z ,M), has the set Z of all integers as the vertex set, and edges ij whenever ji À jj 2 M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z ,M) with clique size at least jMj, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx [22] on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture [26] (also known as the ''lonely runner conjecture'' [1]).

show abstract

“…The parameter rtðTÞ has been studied by Rabinowitz and Proulx [22] and by Griggs and Liu [14]. It was noted in Ref.…”

Section: Almost Difference Closed Sets Of Type A2mentioning

confidence: 98%

“…It was noted in Ref. [14] that ðMÞ ¼ rtðTÞ, if T ¼ M [ f0g. Some known results used in this section are originally given in terms of rtðTÞ.…”

Section: Almost Difference Closed Sets Of Type A2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

“…By some result in Ref. [14] and [2], we may assume that S is periodic, i.e., there is an integer n such that for any k ! 0, S \ fkn; kn þ 1; .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

Liu

Zhu

2004

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…The T -spans for different families of graphs and for different T -sets have been studied extensively (see [3,4,5,10,11,12,14,15,18]). It is known [3,10] that if T is an r-initial set, that is, T = {0, 1, 2, .…”

Section: F (X) − F (Y)| / ∈ T If Xy ∈ E(g)mentioning

confidence: 99%

Minimum Span of No-Hole (r+1)-Distant Colorings

Chang¹,

Juan²,

Liu³

2001

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

Abstract. Given a nonnegative integer r, a no-hole (r + 1)-distant coloring, called Nr-coloring, of a graph G is a function that assigns a nonnegative integer (color) to each vertex such that the separation of the colors of any pair of adjacent vertices is greater than r, and the set of the colors used must be consecutive. Given r and G, the minimum Nr-span of G, nsp r (G), is the minimum difference of the largest and the smallest colors used in an Nr-coloring of G if there exists one; otherwise, define nsp r (G) = ∞. The values of nsp 1 (G) (r = 1) for bipartite graphs are given by Roberts [Math. Comput. Modelling, 17 (1993), pp. 139-144]. Given r ≥ 2, we determine the values of nsp r (G) for all bipartite graph with at least r − 2 isolated vertices. This leads to complete solutions of nsp 2 (G) for bipartite graphs.

show abstract

Distance graphs with missing multiples in the distance sets

Liu

Zhu

1999

J. Graph Theory

View full text Add to dashboard Cite

Given positive integers m, k and s with m > ks, let D m,k,s represent the set {1, 2, • • • , m} − {k, 2k, • • • , sk}. The distance graph G(Z, D m,k,s ) has as vertex set all integers Z and edges connecting i and j whenever |i − j| ∈ D m,k,s . The chromatic number and the fractional chromatic number of G(Z, D m,k,s ) are denoted by χ(Z, D m,k,s ) and χ f (Z, D m,k,s ), respectively. For s = 1, χ(Z, D m,k,1 ) was studied by Eggleton, Erdős and Skilton [6], Kemnitz and Kolberg [12], and Liu [13], and was solved lately by Chang, Liu and Zhu [2] who also determined χ f (Z, D m,k,1 ) for any m and k. This article extends the study of χ(Z, D m,k,s ) and χ f (Z, D m,k,s ) to general values of s. We proveThe latter result provides a good lower bound for χ(Z, D m,k,s ). A general upper bound for χ(Z, D m,k,s ) is found. We prove the upper bound can be improved to (m + sk + 1)/(s + 1) + 1 for some values of m, k and s. In particular, when s + 1 is prime, χ(Z, D m,k,s ) is either (m + sk + 1)/(s + 1) or (m + sk + 1)/(s + 1) + 1. By using a special coloring method called the pre-coloring method, many distance graphs G(Z, D m,k,s ) are classified into * Supported in part by the National Science Foundation under grant DMS 9805945. † Supported in part by the National Science Council under grant NSC87-2115-M110-004.these two possible values of χ(Z, D m,k,s ). Moreover, complete solutions of χ(Z, D m,k,s ) for several families are determined including the case s = 1 (solved in [2]), the case s = 2, the case (k, s + 1) = 1, and the case that k is a power of a prime.

show abstract

The channel assignment problem for mutually adjacent sites

Cited by 30 publications

References 6 publications

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

Minimum Span of No-Hole (r+1)-Distant Colorings

Distance graphs with missing multiples in the distance sets

Contact Info

Product

Resources

About