Planar Graphs with Homomorphisms to the 9-cycle
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910318664351744 |
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| author | Cranston, Daniel W. Li, Jiaao Wang, Zhouningxin Wei, Chunyan |
| author_facet | Cranston, Daniel W. Li, Jiaao Wang, Zhouningxin Wei, Chunyan |
| contents | We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd cycle $C_{2k+1}$. The $k=1$ case is the well-known Grötzsch's $3$-coloring theorem. For general $k$, in 2013 Lovász, Thomassen, Wu, and Zhang showed that it suffices to have odd-girth at least $6k+1$. Improvements are known for $C_5$ and $C_7$ in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For $C_9$ we improve this hypothesis by showing that it suffices to have odd-girth 23. Our main tool is a variation on the potential method applied to modular orientations. This allows more flexibility when seeking reducible configurations. The same techniques also prove some results on circular coloring of signed planar graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Planar Graphs with Homomorphisms to the 9-cycle Cranston, Daniel W. Li, Jiaao Wang, Zhouningxin Wei, Chunyan Combinatorics 05C15, 05C10, 05C21, 05C22 We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd cycle $C_{2k+1}$. The $k=1$ case is the well-known Grötzsch's $3$-coloring theorem. For general $k$, in 2013 Lovász, Thomassen, Wu, and Zhang showed that it suffices to have odd-girth at least $6k+1$. Improvements are known for $C_5$ and $C_7$ in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For $C_9$ we improve this hypothesis by showing that it suffices to have odd-girth 23. Our main tool is a variation on the potential method applied to modular orientations. This allows more flexibility when seeking reducible configurations. The same techniques also prove some results on circular coloring of signed planar graphs. |
| title | Planar Graphs with Homomorphisms to the 9-cycle |
| topic | Combinatorics 05C15, 05C10, 05C21, 05C22 |
| url | https://arxiv.org/abs/2402.02689 |