Cycle decompositions of pathwidth‐6 graphs

Hajós' 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 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 .