On-line coloring of $I_s$-free graphs
An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed...
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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/3472/pdf |
| _version_ | 1827324158110859264 |
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| author | Iwona Cieslik Marcin Kozik Piotr Micek |
| author_facet | Iwona Cieslik Marcin Kozik Piotr Micek |
| author_sort | Iwona Cieslik |
| collection | DOAJ |
| description | An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound. |
| first_indexed | 2024-04-25T02:07:35Z |
| format | Article |
| id | doaj.art-b58ecb44eac14518b6e1fe8237ae2450 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:07:35Z |
| publishDate | 2005-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-b58ecb44eac14518b6e1fe8237ae24502024-03-07T14:32:21ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502005-01-01DMTCS Proceedings vol. AF,...Proceedings10.46298/dmtcs.34723472On-line coloring of $I_s$-free graphsIwona Cieslik0Marcin Kozik1Piotr Micek2Algorithmics Research GroupAlgorithmics Research GroupAlgorithmics Research GroupAn on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.https://dmtcs.episciences.org/3472/pdfplanar graphon-line algorithmgraph coloring[info.info-lo] computer science [cs]/logic in computer science [cs.lo][info.info-sc] computer science [cs]/symbolic computation [cs.sc][info.info-hc] computer science [cs]/human-computer interaction [cs.hc] |
| spellingShingle | Iwona Cieslik Marcin Kozik Piotr Micek On-line coloring of $I_s$-free graphs Discrete Mathematics & Theoretical Computer Science planar graph on-line algorithm graph coloring [info.info-lo] computer science [cs]/logic in computer science [cs.lo] [info.info-sc] computer science [cs]/symbolic computation [cs.sc] [info.info-hc] computer science [cs]/human-computer interaction [cs.hc] |
| title | On-line coloring of $I_s$-free graphs |
| title_full | On-line coloring of $I_s$-free graphs |
| title_fullStr | On-line coloring of $I_s$-free graphs |
| title_full_unstemmed | On-line coloring of $I_s$-free graphs |
| title_short | On-line coloring of $I_s$-free graphs |
| title_sort | on line coloring of i s free graphs |
| topic | planar graph on-line algorithm graph coloring [info.info-lo] computer science [cs]/logic in computer science [cs.lo] [info.info-sc] computer science [cs]/symbolic computation [cs.sc] [info.info-hc] computer science [cs]/human-computer interaction [cs.hc] |
| url | https://dmtcs.episciences.org/3472/pdf |
| work_keys_str_mv | AT iwonacieslik onlinecoloringofisfreegraphs AT marcinkozik onlinecoloringofisfreegraphs AT piotrmicek onlinecoloringofisfreegraphs |