The Chvátal-Erdős theorem (1972) states that a graph with connectivity at least as large as its independence number is Hamiltonian — it contains a cycle visiting every vertex exactly once. The condition balances two opposing forces: high connectivity provides many paths between vertices; low independence number prevents isolated clumps that cycles must detour around.
When the independence number exceeds the connectivity by one, the graph may not be Hamiltonian — you can't always close a single cycle through all vertices. But how far from Hamiltonian can it be?
Chen et al. (arXiv:2603.12776) show the answer is: barely. If a connected graph on at least 3κ+3 vertices has independence number at most κ+1 (one more than its connectivity κ), it contains a 2-factor with at most two components — a spanning subgraph where every vertex has degree exactly 2, consisting of at most two disjoint cycles that together cover all vertices. With one known exceptional family excluded, the graph that fails to be Hamiltonian splits into at most two cycles.
The result is tight: examples exist where two components are necessary and where the vertex bound is needed. The gap between “visits every vertex in one cycle” and “visits every vertex in two cycles” is exactly what happens when the Chvátal-Erdős condition is missed by one. The graph doesn't fragment — it almost closes. The failure of Hamiltonicity is minimal.