On Hamiltonian Cycle Extension in Cubic Hamiltonian Graphs

A Hamiltonian graph \(G\) is said to be \(\ell\)-path-Hamiltonian, where \(\ell\) is a positive integer less than or equal to the order of \(G\), if every path of order \(\ell\) in \(G\) is a subpath of some Hamiltonian cycle in \(G\). The Hamiltonian cycle extension number of \(G\) is the maximum positive integer \(L\) for which \(G\) is \(\ell\)-path-Hamiltonian for every integer \(\ell\) with \(1 \leq \ell \leq L\). Hamiltonian cycle extension numbers are determined for several well-known cubic Hamiltonian graphs. It is shown that if \(G\) is a cubic Hamiltonian graph with girth \(g\), where \(3 \leq g \leq 7\), then \(G\) is \(\ell\)-path-Hamiltonian only if \(1 \leq \ell \leq g\).