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While searching The information System on Graph Classes and their Inclusions, I found several graph classes for which the Hamiltonian Cycle problem is NP-complete while the complexity of Hamiltonian Path problems is NOT known.

Some of those classes are bipartite maximum degree 3 graphs, maximum degree 3 grid graphs, and 2-connected cubic planar graphs.

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bundles/ondemand/components/__styles__/Instructor Message\",\"js/lib/key Mirror\",\"bundles/naptimejs/resources/on Demand Sessions.v1\",\"bundles/naptimejs/resources/on Demand Course Materials.v2\",\"bundles/naptimejs/resources/on Demand Course Schedules.v1\",\"bundles/naptimejs/resources/on Demand Courses Progress.v1\",\"bundles/naptimejs/resources/on Demand Guided Next Steps.v1\",\"bundles/naptimejs/resources/on Demand Course Material Modules.v1\",\"bundles/naptimejs/resources/on Demand Course Material Lessons.v1\",\"bundles/naptimejs/resources/on Demand Course Material Items.v2\",\"bundles/naptimejs/resources/on Demand Guided Items Progress.v1\",\"bundles/next-step/constants\",\"bundles/next-step/lib/build Next Step UIData\",\"bundles/dashboard-2016/components/Course Options Dropdown\",\"bundles/next-step/components/Next Step\",\"bundles/ondemand/utils/progress/utils/compute Weeks To Modules\",\"bundles/ondemand/utils/progress/utils/compute Item Status\",\"bundles/guided-learning-experience/components/Course Progress Bar\",\"bundles/guided-learning-experience/containers/Course Congratulations Banner Container\",\"bundles/ondemand/constants/Prop Types\",\"bundles/ondemand/utils/s12n Upgrade Utils\",\"bundles/coursera-ui/components/extended/Stacked Notification\",\"bundles/naptimejs/resources/user Preferences.v1\",\"bundles/naptimejs/resources/on Demand Specialization Upgrade Eligibilities.v1\",\"bundles/naptimejs/resources/certificate Codes.v1\",\"bundles/enroll/utils/catalog Subscription Certificate Upsell\",\"css!It is unlikely that HP is in $P$ and HC is $NP$-complete for any graph class since there is a Cook reduction from HC to HP which makes at most $O(|E|)$ calls to HP's oracle. Now, the Hamiltonian cycle problem and Hamiltonian path problem are stated to be NP-complete on 2-connected cubic planar graphs.The real question is whether a Karp reduction exist ($HC is NP-complete. However, the computational complexities of HC and HP problems are listed unknown for one problem and NP-complete for the other on circle graphs, triangular grid graphs, and solid grid graphs.Marušič (1982) showed that every connected vertex-transitive graph of order , , , and is Hamiltonian, while Marušič and Parsons (1983) showed that connected vertex-transitive graphs of order and are traceable (Gould, p. Thomassen (1991) conjectured that there are only a finite number of connected nonhamiltonian vertex-transitive graphs, while Babai (1979, 1996) conjectured that there are infinitely many. "Automorphism Groups, Isomorphism, Reconstruction." Ch. There are currently only five known connected nonhamiltonian vertex-transitive graphs, namely the path graph , the Petersen graph , the Coxeter graph , and the triangle-replaced Petersen and Coxeter graphs (the first of these being the cubic vertex-transitive graph Ct66).

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