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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| Vydané v: | Discrete mathematics and theoretical computer science Ročník DMTCS Proceedings vol. AF,...; číslo Proceedings; s. 61 - 68 |
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| Hlavní autori: | , , |
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| Jazyk: | English |
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DMTCS
01.01.2005
Discrete Mathematics and Theoretical Computer Science Discrete Mathematics & Theoretical Computer Science |
| Edícia: | DMTCS Proceedings |
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| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Micek, Piotr Cieslik, Iwona Kozik, Marcin |
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| Keywords | on-line algorithm graph coloring planar graph |
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| SubjectTerms | [info.info-hc] computer science [cs]/human-computer interaction [cs.hc] [info.info-lo] computer science [cs]/logic in computer science [cs.lo] [info.info-sc] computer science [cs]/symbolic computation [cs.sc] Computer Science graph coloring Human-Computer Interaction Logic in Computer Science on-line algorithm planar graph Symbolic Computation |
| Title | On-line coloring of $I_s$-free graphs |
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