Partitioning triangle-free planar graphs into a forest and a linear forest
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912725893906432 |
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| author | Liu, Guanwu Xu, Rongxing |
| author_facet | Liu, Guanwu Xu, Rongxing |
| contents | Raspaud and Wang conjectured that every triangle-free planar graph can be vertex-partitioned into an independent set and a forest. Independently, Kawarabayashi and Thomassen also remarked that this might be true, after providing another proof of a result of Borodin and Glebov, showing this result for planar graphs of girth~5. Subsequently, Dross, Montassier, and Pinlou raised the same question and proved that every triangle-free planar graph can be partitioned into a forest and another forest of maximum degree~5. More recently, Feghali and Šámal improved this bound on the maximum degree to~3. In this note, we further improve the result by showing that every triangle-free planar graph can be partitioned into a forest and a linear forest, that is, a forest of maximum degree~2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partitioning triangle-free planar graphs into a forest and a linear forest Liu, Guanwu Xu, Rongxing Combinatorics Raspaud and Wang conjectured that every triangle-free planar graph can be vertex-partitioned into an independent set and a forest. Independently, Kawarabayashi and Thomassen also remarked that this might be true, after providing another proof of a result of Borodin and Glebov, showing this result for planar graphs of girth~5. Subsequently, Dross, Montassier, and Pinlou raised the same question and proved that every triangle-free planar graph can be partitioned into a forest and another forest of maximum degree~5. More recently, Feghali and Šámal improved this bound on the maximum degree to~3. In this note, we further improve the result by showing that every triangle-free planar graph can be partitioned into a forest and a linear forest, that is, a forest of maximum degree~2. |
| title | Partitioning triangle-free planar graphs into a forest and a linear forest |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.02038 |