Proof of a conjecture of Voss on bridges of longest cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911142536806400 |
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| author | Ma, Jie Xu, Rongxing |
| author_facet | Ma, Jie Xu, Rongxing |
| contents | Bridges are a classical concept in structural graph theory and play a fundamental role in the study of cycles. A conjecture of Voss from 1991 asserts that if disjoint bridges $B_1, B_2, \ldots, B_k$ of a longest cycle $L$ in a $2$-connected graph overlap in a tree-like manner (i.e., induce a tree in the {\it overlap graph} of $L$), then the total {\it length} of these bridges is at most half the length of $L$. Voss established this for $k \leq 3$ and used it as a key tool in his 1991 monograph on cycles and bridges. In this paper, we confirm the conjecture in full via a reduction to a cycle covering problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture of Voss on bridges of longest cycles Ma, Jie Xu, Rongxing Combinatorics Bridges are a classical concept in structural graph theory and play a fundamental role in the study of cycles. A conjecture of Voss from 1991 asserts that if disjoint bridges $B_1, B_2, \ldots, B_k$ of a longest cycle $L$ in a $2$-connected graph overlap in a tree-like manner (i.e., induce a tree in the {\it overlap graph} of $L$), then the total {\it length} of these bridges is at most half the length of $L$. Voss established this for $k \leq 3$ and used it as a key tool in his 1991 monograph on cycles and bridges. In this paper, we confirm the conjecture in full via a reduction to a cycle covering problem. |
| title | Proof of a conjecture of Voss on bridges of longest cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.06345 |