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arXiv · 2507.21565

Solid bricks that every $b$-invariant edge is solitary

Yipei Zhang·Xiumei Wang
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Abstract

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 $W_n$.

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Yipei Zhang, Xiumei Wang. 2025-07-29. Solid bricks that every $b$-invariant edge is solitary. https://arxiv.org/abs/2507.21565

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