DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 171-174

DMTCS

2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

Stefan Felsner (ed.)

DMTCS Conference Volume AE (2005), pp. 171-174


author: Javier Barajas and Oriol Serra
title: Distance graphs with maximum chromatic number
keywords: Distance graphs, 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 ≡0
mod
2
and
b ≡1
mod
2
. This confirms Zhu's conjecture for
|D|=4
.
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reference: Javier Barajas and Oriol Serra (2005), Distance graphs with maximum chromatic number, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 171-174
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