Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

TL;DR

This paper proves that Hamiltonian path, cycle, and linkage problems are solvable in polynomial time for graphs with bounded independence number, resolving open complexity questions for these classes.

Contribution

It establishes polynomial-time algorithms for Hamiltonian problems in graphs with bounded independence number, including a new Hamiltonian-ll-Linkage problem.

Findings

Hamiltonian path and cycle are polynomial-time solvable in graphs with bounded independence number.

Hamiltonian-ll-Linkage problem is polynomial-time solvable for graphs with bounded independence number.

Addresses open complexity questions for special graph classes.

Abstract

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to the first problems shown to be computationally hard when the theory of NP-completeness was being developed. A lot of research has been devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems for special graph classes, yet only a handful of positive results are known. The complexities of both of these problems have been open even for $4K_1$-free graphs, i.e., graphs of independence number at most $3$. We answer this question in the general setting of graphs of bounded independence number. We also consider a newly introduced problem called \emph{Hamiltonian-$\ell$-Linkage} which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given $\ell$ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For $\ell=1$, Hamiltonian-1-Linkage asks for existence of a Hamiltonian path connecting a given pair of vertices. Our main result reads that for every pair of integers $k$ and $\ell$, the Hamiltonian-$\ell$-Linkage problem is polynomial time solvable for graphs of independence number not exceeding $k$.