Distance graphs with maximum chromatic number
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 o...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2005-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3391/pdf |
| Summary: | 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 $|D|=4$. |
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| ISSN: | 1365-8050 |