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.

A graph is called t-perfect if its stable set polytope is fully described by non-negativity, edge and odd-cycle constraints. We characterise P5-free t-perfect graphs in terms of forbidden t-minors. Moreover, we show that P5-free t-perfect graphs can always be coloured with three colours, and that they can be recognised in polynomial time.

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