Solid bricks that every $b$-invariant edge is solitary
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911081639706624 |
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| author | Zhang, Yipei Wang, Xiumei |
| author_facet | Zhang, Yipei Wang, Xiumei |
| contents | A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. A brick $G$ is solid if for any two vertex disjoint odd cycles $C_1$ and $C_2$ of $G$, $G-(V(C_1)\cup V(C_2))$ has no perfect matching. Lucchesi and Murty proposed a problem concerning the characterization of bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. In this paper, we show that for a solid brick $G$ of order $n$ that is distinct from $K_4$, every $b$-invariant edge of $G$ is solitary if and only if $G$ is a wheel $W_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solid bricks that every $b$-invariant edge is solitary Zhang, Yipei Wang, Xiumei Combinatorics A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. A brick $G$ is solid if for any two vertex disjoint odd cycles $C_1$ and $C_2$ of $G$, $G-(V(C_1)\cup V(C_2))$ has no perfect matching. Lucchesi and Murty proposed a problem concerning the characterization of bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. In this paper, we show that for a solid brick $G$ of order $n$ that is distinct from $K_4$, every $b$-invariant edge of $G$ is solitary if and only if $G$ is a wheel $W_n$. |
| title | Solid bricks that every $b$-invariant edge is solitary |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.21565 |