A Tight Lower Bound on Cubic Vertices and Upper Bounds on Thin and Non-thin edges in Planar Braces

TL;DR

This paper establishes a tight lower bound on the number of cubic vertices in planar braces and provides upper bounds on thin and non-thin edges, advancing understanding of their structural properties.

Contribution

It proves that no planar brace exists without cubic vertices and offers precise bounds on the number of cubic vertices and edges in such graphs.

Findings

No planar brace is free of cubic vertices.

Established tight lower bounds on cubic vertices in planar braces.

Provided upper bounds on thin and non-thin edges in planar braces.

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.