The sharp upper bounds on the maximum degree and vertex-connectivity of claw-free 1-planar graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866909870304788480 |
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| author | Zhang, Licheng Ouyang, Zhangdong Huang, Yuanqiu |
| author_facet | Zhang, Licheng Ouyang, Zhangdong Huang, Yuanqiu |
| contents | The complete bipartite graph $K_{1,3}$ is called a claw. The properties of claw-free graphs have attracted considerable attention, with research on claw-free planar graphs tracing back to Plummer's work in 1989. In this paper, we extend this line of research by establishing some fundamental results for claw-free 1-planar graphs, focusing on upper bounds for maximum degree and vertex-connectivity. We show that the maximum degree of claw-free 1-planar graphs is at most 10, and the bound is sharp. Furthermore, we show that for 6-connected 1-planar graphs and optimal 1-planar graphs under the constraint of forbidding induced claws, the maximum degree has the better upper bound 8. Finally, we show that every 7-connected 1-planar graph contains an induced claw, thereby implying that the vertex-connectivity of claw-free 1-planar graphs is at most 6. For a better comparison, we also refine some known results by Plummer on claw-free planar graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The sharp upper bounds on the maximum degree and vertex-connectivity of claw-free 1-planar graphs Zhang, Licheng Ouyang, Zhangdong Huang, Yuanqiu Combinatorics 05C10, 05C40, 05C62, 05C69 The complete bipartite graph $K_{1,3}$ is called a claw. The properties of claw-free graphs have attracted considerable attention, with research on claw-free planar graphs tracing back to Plummer's work in 1989. In this paper, we extend this line of research by establishing some fundamental results for claw-free 1-planar graphs, focusing on upper bounds for maximum degree and vertex-connectivity. We show that the maximum degree of claw-free 1-planar graphs is at most 10, and the bound is sharp. Furthermore, we show that for 6-connected 1-planar graphs and optimal 1-planar graphs under the constraint of forbidding induced claws, the maximum degree has the better upper bound 8. Finally, we show that every 7-connected 1-planar graph contains an induced claw, thereby implying that the vertex-connectivity of claw-free 1-planar graphs is at most 6. For a better comparison, we also refine some known results by Plummer on claw-free planar graphs. |
| title | The sharp upper bounds on the maximum degree and vertex-connectivity of claw-free 1-planar graphs |
| topic | Combinatorics 05C10, 05C40, 05C62, 05C69 |
| url | https://arxiv.org/abs/2501.15124 |