# Cycle decompositions of pathwidth-6 graphs

## TL;DR

This paper proves Hajós conjecture for Eulerian graphs with pathwidth at most 6 by developing new cycle decomposition techniques, confirming the conjecture for this class and supporting the small cycle double cover conjecture.

## Contribution

It introduces novel techniques for cycle decompositions that work on common neighborhoods of degree-6 vertices, verifying Hajós conjecture for a new graph class.

## Key findings

- Hajós conjecture holds for Eulerian graphs with pathwidth ≤ 6
- New cycle decomposition methods focus on neighborhoods of degree-6 vertices
- Supports the small cycle double cover conjecture for these graphs

## Abstract

Haj\'os conjecture asserts that a simple Eulerian graph on n vertices can be decomposed into at most (n - 1)/2 cycles. The conjecture is only proved for graph classes in which every element contains vertices of degree 2 or 4. We develop new techniques to construct cycle decompositions. They work on the common neighbourhood of two degree-6 vertices. With these techniques we find structures that cannot occur in a minimal counterexample to Haj\'os conjecture and verify the conjecture for Eulerian graphs of pathwidth at most 6. This implies that these graphs satisfy the small cycle double cover conjecture.