Algorithmic Extensions of Dirac's Theorem

TL;DR

The paper extends Dirac's long-cycle theorem into the algorithmic realm for 2-connected graphs by introducing an above-guarantee parameterization and two main decompositions, Dirac and Erdős–Gallai, to guide cycle growth. It proves that Long Dirac Cycle and Long Erdős–Gallai Path are fixed-parameter tractable with single-exponential dependence on the combined parameter $k+|B|$, yielding running times like $2^{\mathcal{O}(k+|B|)}\cdot n^{\mathcal{O}(1)}$. It also develops a vertex-cover–above–degree framework and a small-separator strategy to obtain almost-Hamiltonian results, culminating in a robust algorithmic toolkit for finding long cycles and paths under near-Dirac conditions. The results are ETH-tight and address several open questions in parameterized longest-cycle problems, while offering modular, realizable algorithms for practitioners studying Hamiltonicity in dense graphs. The work thus bridges classical extremal graph theory with modern algorithmic techniques, enabling efficient decision and constructive procedures for long cycles in graphs with high minimum degree compared to a small exceptional set.

Abstract

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 $δ\geq 2$ contains a cycle with at least $\min\{2δ,n\}$ vertices. In particular, if $δ\geq n/2$, 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 $2δ+1$. 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 $\min\{2δ+k,n\}$? (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 $\min\{2δ+1,n\}$ 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 $δ$, then deciding whether $G$ has a cycle of length at least $\min\{2δ+k, n\}$ 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.

Figures (15)

• Figure 1: The main steps and connections in the proof of Theorem \ref{['theorem:main']}.
• Figure 2: Structure of the paths $P_1$, $P_2$, $Q_1$, and $Q_2$. a) A dashed line shows path $P_1$, a dotted line indicates $P_2$, and solid lines are used for $Q_1$ and $Q_2$. b) Solid lines indicate paths $R_1$, $R_2$, $S_1$, and $S_2$. The thin lines are used for $R_1$ and $R_2$, while the thick lines for $S_1$ and $S_2$. The choice of $w$ is demonstrated in c).
• Figure 3: A schematic example of an Erdős-Gallai decomposition for a path. The components are denoted by their respective types in the decomposition, R0 denotes components consisting entirely of vertices from $B$ (marked by light gray). The four Erdős-Gallai components are marked by thick blue borders.
• Figure 4: Construction of a long path in $G'$ when $P'$ is shorter than $\delta(G-B)$.
• Figure 5: Constructing a path entering an Erdős-Gallai component in Case 1. The path $P$ is highlighted red. The constructed path is thick blue.

Theorems & Definitions (110)

• Theorem 1: Dirac Dirac52
• Theorem 2: Dirac Dirac52
• Theorem 3: Main Theorem
• Corollary 1
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 1: Egawa, Glas, and Locke Egawa1991
• Theorem 8: Dirac Dirac52