DeepAI

# Algorithmic Extensions of Dirac's Theorem

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≥ 2 contains a cycle with at least min{2δ,n} vertices. In particular, if δ≥ 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≥ 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^𝒪(k)· n^𝒪(1). The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own.

• 51 publications
• 46 publications
• 9 publications
• 20 publications
02/07/2019

### Going Far From Degeneracy

An undirected graph G is d-degenerate if every subgraph of G has a verte...
02/07/2022

### Longest Cycle above Erdős-Gallai Bound

In 1959, Erdős and Gallai proved that every graph G with average vertex ...
02/05/2019

### Hamiltonicity below Dirac's condition

Dirac's theorem (1952) is a classical result of graph theory, stating th...
08/03/2022

### Finding a Lower Bound for k-Unbounded Hamiltonian Cycles

Methods to determine the existence of Hamiltonian Cycles in graphs have ...
04/15/2020

### An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the...
03/12/2023

### Abstract Voronoi-like Graphs: Extending Delaunay's Theorem and Applications

Any system of bisectors (in the sense of abstract Voronoi diagrams) defi...
05/09/2022

### Repeated Averages on Graphs

Sourav Chatterjee, Persi Diaconis, Allan Sly and Lingfu Zhang, prompted ...