A Polynomial-Time Algorithm to Determine (Almost) Hamiltonicity of Dense Regular Graphs

We give a polynomial-time algorithm for detecting very long cycles in dense regular graphs. Specifically, we show that, given $\alpha \in (0,1)$, there exists $c=c(\alpha)$ such that the following holds: is $D$-regular graph $G$ on $n$ vertices with $D\geq \alpha n$, determines whether contains cycle at least $n - c$ vertices. The problem becomes NP-complete if drop either density or regularity condition. combines tools from extremal theory and spectral partitioning as well some further algorithmic ingredients.