Removable edges in cubic matching covered graphs

TL;DR

This work studies removable edges in cubic matching-covered graphs, focusing on cubic bricks as fundamental building blocks. It proves that every cubic brick $G$ with $G\neq K_4$ and $G\neq \overline{C_6}$ contains a matching of size at least $|V(G)|/8$ whose edges are removable, with the bound met by the graph $R_8$. The authors develop tools such as splicing and $\Delta$-replacements, and employ an induction on the vertex count to construct large removable-edge matchings. The results deepen understanding of ear decompositions and brick structure, highlighting how local removability scales to large cubic bricks, and noting that the bound is tight in at least some cases like $R_8$.

Abstract

{ An edge $e$ in a matching covered graph $G$ is {\em removable} if $G-e$ is matching covered, which was introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A {\it brick}} is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Improving Lovász's result, Carvalho et al. [Ear decompositions of matching covered graphs, {\em Combinatorica}, 19(2):151-174, 1999] showed that each brick other than $K_4$ and $\overline{C_6}$ has $Δ-2$ removable edges, where $Δ$ is the maximum degree of $G$. In this paper, we show that every cubic brick $G$ other than $K_4$ and $\overline{C_6}$ has a matching of size at least $|V(G)|/8$, each edge of which is removable in $G$.

Figures (6)

• Figure 1: The bold edges represent the removable edges.
• Figure 2: The bold edges represent the removable edges.
• Figure 3: $G[Y_2]$ when $|\{x_2,y_2,z_2\}\cap S_2|= 2$ and $|\{x_2,y_2,z_2\}\cap S_1|=0$.
• Figure 4: $G[Y_2]$ when $|\{x_2,y_2,z_2\}\cap S_2|= 1$ and $\{x_2,y_2\}\cap S_1=\emptyset$.
• Figure 5: $G[Y_2]$ when $|\{x_2,y_2,z_2\}\cap S_2|= 1$ and $|\{x_2,y_2\}\cap S_1|=1$.

Theorems & Definitions (32)

• Theorem 1.1
• Lemma 2.1
• Proposition 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8