Lovász–Woodall conjecture

Many results in graph theory, starting with Menger's theorem, guarantee the existence of paths or cycles in a k-connected graph. For 2-connected graphs, Menger's theorem is equivalent to the statement that any two vertices lie on a common cycle. A theorem of G. A. Dirac generalizes this claim: if a graph is k-connected for k ≥ 2, then for every set of k vertices in the graph there is a cycle that passes through all the vertices in the set.[3]

Another corollary of Menger's theorem is that in 2-connected graphs, any two edges lie on a common cycle. The proof, however, does not generalize to the corresponding statement for k edges in a k-connected graph; rather, Menger's theorem can be used to show that in a k-connected graph, given any 2 edges and k-2 vertices, there is a cycle passing through all of them.[4]

There is one obstacle to the stronger claim that in a k-connected graph G, given any set L of k edges, there should be a cycle containing L. Suppose that the edges in L form an edge cut: the vertices of G can be separated into two sets A and B such that the edges in L all join a vertex in A to a vertex in B, and are the only edges to do so. Then any cycle in G can only use an even number of edges of L: it must cross from A to B and from B back to A an equal number of times. If k is odd, this means that no cycle can contain all of L.

The Lovász–Woodall conjecture states that this is the only obstacle: given any set L of k edges, there is a cycle containing L, except in the case that k is odd and L is an edge cut.

Woodall proposed the conjecture as one of several possible statements[2] that would imply a conjecture made by Claude Berge: given a k-connected graph G with independence number α(G), and any subgraph F of G with at most k-α(G) edges whose components are all paths, G has a Hamiltonian cycle containing F.[5] In 1982, Roland Häggkvist and Carsten Thomassen proved Berge's conjecture by proving one of the weaker statements proposed by Woodall.[6]

As mentioned above, the k = 2 case of the Lovász–Woodall conjecture follows from Menger's theorem. The k = 3 case was given as an exercise by Lovász.[7] After the conjecture was made, it was proven for k = 4 by Péter L. Erdős and E. Győri[8] and independently by Michael V. Lomonosov.[9], and for k = 5 by Daniel P. Sanders.[10]

Other partial progress toward the conjecture has included versions of the result with a stronger assumption on connectivity. Woodall's paper[2] included a proof that the conclusion of the conjecture holds if G is (2k-2)-connected, and in 1977, Thomassen proved that the conjecture holds if G is (3k-1)/2-connected.[11] In 1982, Häggkvist and Thomassen proved that the conjecture holds if G is (k+1)-connected.[6]

In 2002, Ken-ichi Kawarabayashi proved that under the hypotheses of the conjecture, L is either contained in a cycle of G or in two disjoint cycles.[7]

In two publications in 2002[7] and 2008,[12] Kawarabayashi claimed to have a proof on the conjecture, giving an outline for the proof and leaving several steps to future publications, but the full proof has not been published since.