Coloring Grids Avoiding Bicolored Paths
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908905912664064 |
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| author | Keskinkilic, Derman Ozkahya, Lale |
| author_facet | Keskinkilic, Derman Ozkahya, Lale |
| contents | The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Grünbaum (1973) together with the acyclic coloring of graphs, where bicolored cycles are avoided. In this paper, we study a generalization of this problem, by considering proper vertex coloring on graphs forbidding bicolored paths of a fixed length, which was initially discussed by Alon, McDiarmid, and Reed (1991). Here, we study this problem on products of two paths. We show that at least 4 colors are needed to properly color the product of paths, $P_m\square P_n$, avoiding a bicolored $P_k,$ unless $n<k-2$ or $m<k-2.$ With this result, the above question is settled for all $k$ on 2-dimensional grids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12919 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coloring Grids Avoiding Bicolored Paths Keskinkilic, Derman Ozkahya, Lale Combinatorics Discrete Mathematics The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Grünbaum (1973) together with the acyclic coloring of graphs, where bicolored cycles are avoided. In this paper, we study a generalization of this problem, by considering proper vertex coloring on graphs forbidding bicolored paths of a fixed length, which was initially discussed by Alon, McDiarmid, and Reed (1991). Here, we study this problem on products of two paths. We show that at least 4 colors are needed to properly color the product of paths, $P_m\square P_n$, avoiding a bicolored $P_k,$ unless $n<k-2$ or $m<k-2.$ With this result, the above question is settled for all $k$ on 2-dimensional grids. |
| title | Coloring Grids Avoiding Bicolored Paths |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2312.12919 |