The Lovász conjecture, posed in 1969, asks whether every connected vertex-transitive graph has a Hamiltonian path — a path that visits every vertex once. This new paper does not settle that conjecture. But it proves a much stronger lower bound than was previously known: for every small number ε>0 and every sufficiently large connected vertex-transitive graph with n vertices, the graph contains a cycle whose length is at least n^{1−ε}. In other words, the authors show the longest cycle has size n^{1-o(1)}, improving the recent n^{2/3-o(1)} bound.

The proof starts by using a structure theorem of Tessera and Tointon to break the graph into parts. Each part has small diameter in the original graph, and the partition is compatible with all graph symmetries (automorphisms). The authors then form a quotient graph Q whose vertices are the parts; Q is itself vertex-transitive and has a spanning tree whose nodes have degree at most three. From this global picture they handle two different cases depending on the sizes of the parts.

When the parts are large, the authors repeatedly walk around the low-degree spanning tree and try to move between parts by short paths. They pick many short paths at random and use the Lovász local lemma — a probabilistic tool that controls rare bad events — to avoid unwanted overlaps. This lets them stitch many short pieces together into one very long simple path in the original graph.

When the parts are small, there are many parts, so a long path in the quotient Q already gives a long path in the original graph (each step in Q lifts to at least one vertex). To get such a long path in Q the authors bring in group structure provided by the Tessera–Tointon theorem and apply Babai’s contraction lemma. This reduces the problem to Cayley graphs of nilpotent groups with a bounded number of generators and bounded nilpotency class. The paper proves that such Cayley graphs on m vertices contain paths of length m^{1-o(1)}, which yields the near-linear cycle when lifted back.