A Tight Lower Bound on Cubic Vertices and Upper Bounds on Thin and Non-thin edges in Planar Braces
Koustav De
TL;DR
The paper proves that any planar brace must contain at least eight cubic vertices, i.e., $n_3 \ge 8$, by combining the brace degree-sum bound $2m \ge 3n_3 + 4(n - n_3)$ with the planar bipartite edge bound $m \le 2n - 4$, and shows equality is achieved by the cube graph $Q_3$. It further derives sharp upper bounds on edge types in planar braces: $|\mathcal{E}_T| \le n - 19$ and $|\mathcal{E}_{NT}| \le n - 9$ (nonthin edges with endpoints in $S_1$), while also confirming that the subgraph induced by nonthin edges within $S_1$ is a forest, as in the general case. The results force the proportion $k = n_3/n$ to satisfy $k \ge 8/n$, constraining the He–Lu thin-edge lower bound for planar braces, and the work outlines connections to extensions on other surfaces and potential algorithmic applications.
Abstract
For a subset $X$ of the vertex set $\VV(\GG)$ of a graph $\GG$, we denote the set of edges of $\GG$ which have exactly one end in $X$ by $\partial(X)$ and refer to it as the cut of $X$ or edge cut $\partial(X)$. A graph $\GG=(\VV,\EE)$ is called matching covered if $\forall e \in \EE(\GG), ~\exists \text{a perfect matching }M \text{ of }\GG \text{ s. t. } e \in M$. A cut $C$ of a matching covered graph $\GG$ is a separating cut if and only if, given any edge $e$, there is a perfect matching $M_{e}$ of $\GG$ such that $e \in M_{e}$ and $|C \cap M_{e}| = 1$. A cut $C$ in a matching covered graph $\GG$ is a tight cut of $\GG$ if $|C \cap M| = 1$ for every perfect matching $M$ of $\GG$. For, $X, Y \subseteq \VV(\GG)$, we denote the set of edges of $\EE(\GG)$ which have one endpoint in $X$ and the other endpoint in $Y$ by $E[X,Y]$. Let $\partial(X)=E[X,\overline{X}]$ be an edge cut, where $\overline{X}=\VV(\GG) \setminus X$. An edge cut is trivial if $|X|=1$ or $|\overline{X}|=1$. A matching covered graph, which is free of nontrivial tight cuts, is a brace if it is bipartite and is a brick if it is non-bipartite. An edge $e$ in a brace $\GG$ is \emph{thin} if, for every tight cut $\partial(X)$ of $\GG - e$, $|X| \le 3$ or $|\overline{X}| \le 3$. Carvalho, Lucchesi and Murty conjectured that there exists a positive constant $c$ such that every brace $\GG$ has $c|\VV(\GG)|$ thin edges \cite{DBLP:journals/combinatorics/LucchesiCM15}. He and Lu \cite{HE2025153} showed a lower bound of thin edges in a brace in terms of the number of cubic vertices. We asked whether any planar brace exists that does not contain any cubic vertices. We answer negatively by showing that such set of planar braces is empty. We have been able to show a quantitively tight lower bound on the number of cubic vertices in a planar brace. We have proved tight upper bounds of nonthin edges and thin edges in a planar brace.