Path and cycle decompositions of dense graphs

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph n vertices can be decomposed into at most ? 2 ? paths, while a conjecture Hajós states Eulerian ? ? 1 ? cycles. The Erd?s–Gallai O ( ) cycles edges. show if G is sufficiently large with linear minimum degree, then following hold. (i) + o paths. (ii) If Eulerian, it (iii) 3 in addition satisfies weak expansion property, we asymptotically determine required number paths/cycles for each such G. (iv) max { odd , ? } where odd-degree (v) All bounds (i)–(v) are best possible.