Hamiltonicity in connected regular graphs

TL;DR

This paper determines the minimum size of connected regular graphs that lack Hamiltonian cycles or paths, extending classical results and characterizing the smallest non-Hamiltonian regular graphs.

Contribution

It identifies the minimal vertex counts for non-Hamiltonian connected regular graphs and characterizes the smallest such graphs without Hamiltonian cycles.

Findings

Minimum vertices for non-Hamiltonian connected regular graphs identified

Characterization of smallest connected regular graphs without Hamiltonian cycles

Extension of classical Hamiltonicity results in regular graphs

Abstract

In 1980, Jackson proved that every 2-connected $k$-regular graph with at most $3k$ vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected $k$-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected $k$-regular graphs without a Hamiltonian cycle.