Forcibly unicyclic and bicyclic graphic sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909588352139264 |
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| author | Duan, Peiyi Tian, Yingzhi |
| author_facet | Duan, Peiyi Tian, Yingzhi |
| contents | A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical property $\mathcal{P}$, a graphic sequence $D$ is forcibly $\mathcal{P}$ graphic if each graph with degree sequence $D$ has property $\mathcal{P}$. A graph is acyclic if it contains no cycles. A connected acyclic graph is just a tree and has $n-1$ edges. A graph of order $n$ is unicyclic (resp. bicyclic) if it is connected and has $n$ (resp. $n+1$) edges. Bar-Noy, Böhnlein, Peleg and Rawitz [Discrete Mathematics 346 (2023) 113460] characterized forcibly acyclic and forcibly connected acyclic graphic sequences. In this paper, we aim to characterize forcibly unicyclic and forcibly bicyclic graphic sequences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forcibly unicyclic and bicyclic graphic sequences Duan, Peiyi Tian, Yingzhi Combinatorics A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical property $\mathcal{P}$, a graphic sequence $D$ is forcibly $\mathcal{P}$ graphic if each graph with degree sequence $D$ has property $\mathcal{P}$. A graph is acyclic if it contains no cycles. A connected acyclic graph is just a tree and has $n-1$ edges. A graph of order $n$ is unicyclic (resp. bicyclic) if it is connected and has $n$ (resp. $n+1$) edges. Bar-Noy, Böhnlein, Peleg and Rawitz [Discrete Mathematics 346 (2023) 113460] characterized forcibly acyclic and forcibly connected acyclic graphic sequences. In this paper, we aim to characterize forcibly unicyclic and forcibly bicyclic graphic sequences. |
| title | Forcibly unicyclic and bicyclic graphic sequences |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.15596 |