Cycle decompositions of pathwidth-6 graphs

Abstract

Hajós' conjecture asserts that a simple Eulerian graph on (Formula presented.) vertices can be decomposed into at most (Formula presented.) 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 neighborhood of two degree-6 vertices. With these techniques, we find structures that cannot occur in a minimal counterexample to Hajós' 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.