Skip to Main content Skip to Navigation
Conference papers

Distance graphs with maximum chromatic number

Abstract : Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.
Complete list of metadata

Cited literature [11 references]  Display  Hide  Download

https://hal.inria.fr/hal-01184347
Contributor : Coordination Episciences Iam <>
Submitted on : Friday, August 14, 2015 - 11:36:36 AM
Last modification on : Monday, November 16, 2020 - 3:56:03 PM
Long-term archiving on: : Sunday, November 15, 2015 - 10:57:28 AM

File

dmAE0134.pdf
Publisher files allowed on an open archive

Identifiers

  • HAL Id : hal-01184347, version 1

Collections

Citation

Javier Barajas, Oriol Serra. Distance graphs with maximum chromatic number. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.171-174. ⟨hal-01184347⟩

Share

Metrics

Record views

268

Files downloads

909