Circular chromatic number of distance graphs with distance sets of cardinality 3

Zhu, Xuding

doi:10.1002/jgt.10062

Cited by 31 publications

(21 citation statements)

References 16 publications

(33 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…However, we do not know whether there is any other set M attaining this bound. It seems natural to conjecture the following: We note here that for m 3, both Conjectures 5.2 and 5.3 are true [5,7,28].…”

Section: Consequences and Related Problemsmentioning

confidence: 95%

See 1 more Smart Citation

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

Section: Consequences and Related Problemsmentioning

confidence: 95%

“…Partial results on the circular chromatic number and the fractional chromatic number were obtained by Zhu [28].…”

Section: Consequences and Related Problemsmentioning

confidence: 98%

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

“…However, there is a very simple general coloring method that works for many distance graphs: the regular coloring method. This method was used to determine not only the chromatic number but also the circular chromatic number of many distance graphs [2,3,6,14,20,35]. The essence of the regular coloring method is revealed in the proof of Theorem 9 below, which was proved in [31].…”

Section: Proof It Was Proved Inmentioning

confidence: 99%

“…Their motivation was to study the one-dimensional analogue of the well-known plane coloring problem (i.e., finding the minimum number of colors needed to color the plane so that no two points of unit distance are colored the same color). Later on, it was found that the chromatic number and fractional chromatic number of distance graphs are related to many other problems, such as T -colorings [3,24], diophantine approximations [35], density of D-sets [15] and circulant graphs [20], etc.…”

Section: G(z D) For N ≥ 1 We Denote By G(n D) the Subgraph Of G(mentioning

confidence: 99%

Asymptotic Clique Covering Ratios of Distance Graphs

Liu

Zhu

2002

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

Given a finite set D of positive integers, the distance graph G(Z , D) has Z as the vertex set and {i j : |i − j| ∈ D} as the edge set. Given D, the asymptotic clique covering ratio is defined as (Z , D)), where ω(G) is the clique number of G.) This problem turns out to be related to T -colorings and to fractional chromatic number and circular chromatic number of distance graphs. Through such connections, we shall show that the equality S(D) = ω(G(Z , D)) holds for many classes of distance graphs. Moreover, we raise questions regarding other such connections.

show abstract

“…[4] for recent results and many related references. Notably, the chromatic number of G(D) is now determined in [5] for all three-element distance sets D.…”

Section: E={(i J) | I < J J − I ¥ D}mentioning

confidence: 99%