scholarly journals Distance graphs with maximum chromatic number

2008 ◽  
Vol 308 (8) ◽  
pp. 1355-1365 ◽  
Author(s):  
Javier Barajas ◽  
Oriol Serra
Keyword(s):  
Chromatic Number ◽  
Distance Graphs

Chromatic number of distance graphs generated by the sets {2,3,x,y}

2012 ◽  
Vol 25 (4) ◽  
pp. 680-693 ◽  
Author(s):  
Daphne Der-Fen Liu ◽  
Aileen Sutedja
Keyword(s):  
Chromatic Number ◽  
Distance Graphs

scholarly journals On Robust Colorings of Hamming-Distance Graphs

10.37236/6495 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Isaiah Harney ◽  
Heide Gluesing-Luerssen
Keyword(s):  
Chromatic Number ◽  
Hamming Distance ◽  
Explicit Description ◽  
Distance Graphs ◽  
Vertex Set

$H_q(n,d)$ is defined as the graph with vertex set $\mathbb{Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.

scholarly journals On the chromatic number of special distance graphs

1991 ◽  
Vol 97 (1-3) ◽  
pp. 395-397 ◽  
Author(s):  
M. Voigt ◽  
H. Walther
Keyword(s):  
Chromatic Number ◽  
Distance Graphs

scholarly journals Distance graphs with maximum chromatic number

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Javier Barajas ◽  
Oriol Serra
Keyword(s):  
Chromatic Number ◽  
Distance Graph ◽  
Distance Graphs ◽  
Maximum Value ◽  
Finite Set ◽  
International Audience

International audience 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 $|D|=4$.

Sign in / Sign up

Export Citation Format

Share Document