Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number
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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 Hamiltonian-ℓ-Linkage which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given ℓ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For ℓ=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 ℓ, the Hamiltonian-ℓ-Linkage problem is polynomial time solvable for graphs of independence number not exceeding k. We further complement this general polynomial time algorithm by a structural description of obstacles to Hamiltonicity in graphs of independence number at most k for small values of k.
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