One or Two Disjoint Circuits Cover Independent Edges: Lovász–Woodall Conjecture

In this paper, we prove the following theorem: Let L be a set of k independent edges in a k-connected graph G. If k is even or G−L is connected, then there exist one or two disjoint circuits containing all the edges in L. This theorem is the first step in the proof of the conjecture of L. Lovász (1974, Period. Math. Hungar., 82) and D. R. Woodall (1977, J. Combin. Theory Ser. B22, 274–278). In addition, we give the outline of the proof of the conjecture and refer to the forthcoming papers.