In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$ vertices. In particular, if $\delta\geq n/2$, then $G$ is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than $2\delta+1$. Also, there are non-Hamiltonian graphs with all vertices but one of degree at least $n/2$. This prompts naturally to the following algorithmic questions. For $k\geq 1$, (A) How difficult is to decide whether a 2-connected graph contains a cycle of length at least $\min\{2\delta+k,n\}$? (B) How difficult is to decide whether a graph $G$ is Hamiltonian, when at least $n - k$ vertices of $G$ are of degrees at least $n/2-k$? The first question was asked by Fomin, Golovach, Lokshtanov, Panolan, Saurabh, and Zehavi. The second question is due to Jansen, Kozma, and Nederlof. Even for a very special case of $k=1$, the existence of a polynomial-time algorithm deciding whether $G$ contains a cycle of length at least $\min\{2\delta+1,n\}$ was open. We resolve both questions by proving the following algorithmic generalization of Dirac's theorem: If all but $k$ vertices of a $2$-connected graph $G$ are of degree at least $\delta$, then deciding whether $G$ has a cycle of length at least $\min\{2\delta +k, n\}$ can be done in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$. The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own.

翻译：1952年，狄拉克证明了关于最小顶点度数较大的图中长环的如下定理：每个具有最小顶点度数δ≥2的n顶点2-连通图G都包含一个长度至少为min{2δ,n}的环。特别地，若δ≥n/2，则G是哈密顿图。狄拉克定理的证明是构造性的，并由此得到可在多项式时间内计算相应环的算法。狄拉克定理的组合下界在以下意义上是紧的：存在不含长度超过2δ+1的环的2-连通图；此外，存在所有顶点除一个外度数均至少为n/2的非哈密顿图。这自然引出了如下算法问题：对于k≥1，（A）判定一个2-连通图是否包含长度至少为min{2δ+k,n}的环有多困难？（B）当图G中至少有n-k个顶点的度数至少为n/2-k时，判定G是否为哈密顿图有多困难？第一个问题由Fomin、Golovach、Lokshtanov、Panolan、Saurabh和Zehavi提出，第二个问题源于Jansen、Kozma和Nederlof。即使在k=1的特殊情况下，是否存在多项式时间算法判定G是否包含长度至少为min{2δ+1,n}的环仍是一个开放问题。我们通过证明以下狄拉克定理的算法推广解决了这两个问题：若2-连通图G中除k个顶点外所有顶点的度数均至少为δ，则判定G是否具有长度至少为min{2δ+k,n}的环可在时间2^{O(k)}·n^{O(1)}内完成。该算法推广的证明建立在具有独立价值的新图论结果之上。