# Perfect Pseudo-Matchings in cubic graphs

## TL;DR

This paper investigates perfect pseudo-matchings in cubic graphs, focusing on their existence in snarks and their implications for cycle double covers, revealing new classes of graphs with these properties.

## Contribution

It introduces the concept of perfect pseudo-matchings in cubic graphs and demonstrates their presence in various snark classes, linking them to planarity and cycle double covers.

## Key findings

- Well-known snarks contain planarizing perfect pseudo-matchings
- There are at least as many snarks with these matchings as cyclically 5-edge-connected snarks
- Such matchings imply the existence of cycle double covers in the graphs

## Abstract

A perfect pseudo-matching M in a cubic graph G is a spanning subgraph of G such that every component of M is isomorphic to K_2 or to K_1,3. In view of snarks G with dominating cycle C, this is a natural generalization of perfect matchings since G-E(C) is a perfect pseudo-matching. Of special interest are such M where the graph G/M is planar because such G have a cycle double cover. We show that various well known classes of snarks contain planarizing perfect pseudo-matchings, and that there are at least as many snarks with planarizing perfect pseudo-matchings as there are cyclically 5-edge-connected snarks.