Haj\'os' cycle conjecture for small graphs

TL;DR

This paper verifies Hajós' cycle conjecture for all Eulerian graphs with up to twelve nodes using preprocessing, heuristics, and integer programming techniques, advancing understanding of cycle decompositions in small graphs.

Contribution

The paper introduces a computational approach combining preprocessing, heuristics, and integer programming to verify Hajós' conjecture for small Eulerian graphs.

Findings

Hajós' conjecture holds for all Eulerian graphs with up to 12 nodes.

The developed methods effectively verify the conjecture in small graphs.

The approach can potentially be extended to larger graphs with further optimization.

Abstract

Haj\'os' conjecture states that an Eulerian graph of order n can be decomposed into at most (n-1)/2 edge-disjoint cycles. We describe preprocessing steps, heuristics and integer programming techniques that enable us to verify Haj\'os' conjecture for all Eulerian graphs with up to twelve nodes.