# Snarks with special spanning trees

## TL;DR

This paper investigates the cycle lengths in the 2-regular subgraph of cubic snarks that can be decomposed into a spanning tree without degree-two vertices and a 2-regular subgraph, addressing a specific structural question.

## Contribution

It characterizes the possible cycle lengths in the 2-regular subgraph of snarks with a special spanning tree decomposition.

## Key findings

- Identifies restrictions on cycle lengths in the 2-regular subgraph of snarks.
- Provides structural insights into cubic graphs with a hist and their decompositions.
- Answers a specific open question about cycle lengths in these decompositions.

## Abstract

Let $G$ be a cubic graph which has a decomposition into a spanning tree $T$ and a $2$-regular subgraph $C$, i.e. $E(T) \cup E(C) = E(G)$ and $E(T) \cap E(C) = \emptyset$. We provide an answer to the following question: which lengths can the cycles of $C$ have if $G$ is a snark? Note that $T$ is a hist (i.e. a spanning tree without a vertex of degree two) and that every cubic graph with a hist has the above decomposition.