2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 171174
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
and
mod
2
b ≡1
. This confirms Zhu's conjecture for
mod
2
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. 171174 
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