# Nice vertices in cubic graphs

## TL;DR

This paper investigates the number of nice vertices in matching covered cubic graphs, establishing lower bounds for non-bipartite and bipartite cases, and characterizing extremal graphs with minimal nice vertices or pairs.

## Contribution

It provides new lower bounds on the number of nice vertices in 2- and 3-connected cubic graphs and characterizes extremal graphs, including the unique extremal bipartite case.

## Key findings

- Non-bipartite 2-connected cubic graphs have at least 4 nice vertices.
- 3-connected non-bipartite cubic graphs (not K4) have at least 6 nice vertices.
- Connected bipartite cubic graphs have at least 9 nice pairs, with K_{3,3} as the extremal case.

## Abstract

A subgraph $G'$ of a graph $G$ is nice if $G-V(G')$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex $u$ of a cubic graph is nice if $u$ and its neighbors induce a nice subgraph. D. Kr\'{a}l et al. (2010) [9] showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph $G$ is 2-connected, then $G$ has at least 4 nice vertices; if $G$ is 3-connected and $G\neq K_4$, then $G$ has at least 6 nice vertices. We also determine all the corresponding extremal graphs. For a cubic bipartite graph $G$ with bipartition $(A,B)$, a pair of vertices $a\in A$ and $b\in B$ is called a nice pair if $a$ and $b$ together with their neighbors induce a nice subgraph. We show that a connected cubic bipartite graph $G$ is a brace if and only if each pair of vertices in distinct color classes is a nice pair. In general, we prove that $G$ has at least 9 nice pairs of vertices and $K_{3,3}$ is the only extremal graph.