The Odd-Distance Plane Graph

Ardal, Hayri; Maňuch, Ján; Rosenfeld, Moshe; Shelah, Saharon; Stacho, Ladislav

doi:10.1007/s00454-009-9190-2

Cited by 20 publications

(21 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, the unit-distance plane graph is a subgraph of the odd distance graph, thus the chromatic number of the odd-distance plane graph is at least 4. Only recently, we managed to prove that the chromatic number of the odd-distance plane graph is at least 5 [3]. In this paper we also proved that this graph is ℵ 0 -choosable while the odd-distance graph in R 3 is not countably choosable, a marked difference from the unit-distance graph in R 3 .…”

Section: The Odd-distance Graphmentioning

confidence: 71%

Some of my favorite “lesser known” problems

Rosenfeld

2008

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: The Odd-distance Graphmentioning

confidence: 71%

Some of my favorite “lesser known” problems

Rosenfeld

2008

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The recent paper [2] rediscovered some of the above results, namely, List(X 2 (ODD)) = ℵ 0 < List(X 3 (ODD)), where ODD denotes the set of odd natural numbers. The authors of [2] were primarily interested in the fascinating, and apparently hard, conjecture that Chr(X 2 (ODD)) is finite.…”

Section: History a Seminal Results Of Erdős And Hajnal On Infinite Gramentioning

confidence: 96%

“…The authors of [2] were primarily interested in the fascinating, and apparently hard, conjecture that Chr(X 2 (ODD)) is finite.…”

Section: History a Seminal Results Of Erdős And Hajnal On Infinite Gramentioning

confidence: 99%

The List-Chromatic Number of Infinite Graphs Defined on Euclidean Spaces

Komjáth

2009

Discrete Comput Geom

View full text Add to dashboard Cite

If D is a countable set of positive reals, 2 ≤ n < ω, let X n (D) be the graph with the points of R n as vertices where two vertices are joined iff their distance is in D. We determine the list-chromatic number of X n (D) as much as possible.Keywords List-chromatic number · Coloring number · Graphs defined by distances · Euclidean space · Independence proofs If D is a countable set of positive numbers, then let X n (D) denote the n-dimensional D-distance graph. That is, the vertex set is R n , and two points are joined if their distance is in D.In this paper we make some remarks on List(X n (D)) for certain values of D and n, where for a graph X, Chr(X) denotes the chromatic number, Col(X) the coloring number, and List(X) the list-chromatic number of X. The inequality Chr(X) ≤ List(X) ≤ Col(X) immediately connects these notions. In general, Chr(X) and Col(X) can be far away; there are bipartite graphs with large coloring number. Namely, Col(K(κ, κ)) = κ and Col(K(κ, λ)) = κ + when κ < λ are infinite.The results of [8] show that the list-chromatic number can be close to the chromatic number and also to the coloring number, depending on the model of set theory. We determine List(X n (D)) as much as it is possible. Namely, List(X 2 (D)) = ℵ 0 always holds, List(X 3 (D)) is countable if D is finite or is a sequence converging to 0 and is ℵ 1 otherwise, ℵ 1 ≤ List(X n (D)) ≤ c (4 ≤ n < ω) and, depending on the model of set theory, can be either of the two extrema.

show abstract

“…Erdős [6] raised the problem of determining the maximal number of edges in a unitdistance graph on n vertices and this question became known as the Erdős Unit Distance Problem. Erdős and Rosenfeld [2] asked analogous questions for odd distances. An odd-distance graph is a geometric graph in which edges are represented by segments whose length is an odd integer.…”

Section: Introductionmentioning

confidence: 99%