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...
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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 |
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| Online Access: | https://dmtcs.episciences.org/3391/pdf |
| _version_ | 1827324095185813504 |
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| author | Javier Barajas Oriol Serra |
| author_facet | Javier Barajas Oriol Serra |
| author_sort | Javier Barajas |
| collection | DOAJ |
| description | 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 $|D|=4$. |
| first_indexed | 2024-04-25T02:03:16Z |
| format | Article |
| id | doaj.art-c1a6cf8e88c34d3cb0feb6fa59fcfe1f |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:16Z |
| publishDate | 2005-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-c1a6cf8e88c34d3cb0feb6fa59fcfe1f2024-03-07T14:41:15ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502005-01-01DMTCS Proceedings vol. AE,...Proceedings10.46298/dmtcs.33913391Distance graphs with maximum chromatic numberJavier Barajas0Oriol Serra1https://orcid.org/0000-0001-8561-4631Universitat Politècnica de Catalunya [Barcelona]Universitat Politècnica de Catalunya [Barcelona]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 $|D|=4$.https://dmtcs.episciences.org/3391/pdfdistance graphschromatic number[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Javier Barajas Oriol Serra Distance graphs with maximum chromatic number Discrete Mathematics & Theoretical Computer Science distance graphs chromatic number [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Distance graphs with maximum chromatic number |
| title_full | Distance graphs with maximum chromatic number |
| title_fullStr | Distance graphs with maximum chromatic number |
| title_full_unstemmed | Distance graphs with maximum chromatic number |
| title_short | Distance graphs with maximum chromatic number |
| title_sort | distance graphs with maximum chromatic number |
| topic | distance graphs chromatic number [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3391/pdf |
| work_keys_str_mv | AT javierbarajas distancegraphswithmaximumchromaticnumber AT oriolserra distancegraphswithmaximumchromaticnumber |