Negative results on acyclic improper colorings
Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$. We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar gr...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2005-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3441/pdf |
| _version_ | 1797270357282914304 |
|---|---|
| author | Pascal Ochem |
| author_facet | Pascal Ochem |
| author_sort | Pascal Ochem |
| collection | DOAJ |
| description | Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$. We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar graphs. We try to get the most restrictive conditions on the class $\mathcal{C}$, such as having large girth and small maximum degree. In particular, we obtain the $\mathrm{NP}$-completeness of $3$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $4$, and of $4$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $8$. |
| first_indexed | 2024-04-25T02:02:59Z |
| format | Article |
| id | doaj.art-74a3cd9f11e34ad99515d29de8b99219 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:59Z |
| publishDate | 2005-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-74a3cd9f11e34ad99515d29de8b992192024-03-07T14:41:15ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502005-01-01DMTCS Proceedings vol. AE,...Proceedings10.46298/dmtcs.34413441Negative results on acyclic improper coloringsPascal Ochem0https://orcid.org/0000-0001-5504-4586Laboratoire Bordelais de Recherche en InformatiqueRaspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$. We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar graphs. We try to get the most restrictive conditions on the class $\mathcal{C}$, such as having large girth and small maximum degree. In particular, we obtain the $\mathrm{NP}$-completeness of $3$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $4$, and of $4$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $8$.https://dmtcs.episciences.org/3441/pdf$\mathrm{np}$-completenessacyclic coloringsoriented colorings[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Pascal Ochem Negative results on acyclic improper colorings Discrete Mathematics & Theoretical Computer Science $\mathrm{np}$-completeness acyclic colorings oriented colorings [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Negative results on acyclic improper colorings |
| title_full | Negative results on acyclic improper colorings |
| title_fullStr | Negative results on acyclic improper colorings |
| title_full_unstemmed | Negative results on acyclic improper colorings |
| title_short | Negative results on acyclic improper colorings |
| title_sort | negative results on acyclic improper colorings |
| topic | $\mathrm{np}$-completeness acyclic colorings oriented colorings [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3441/pdf |
| work_keys_str_mv | AT pascalochem negativeresultsonacyclicimpropercolorings |