On the number of Hamiltonian cycles in the generalized Petersen graph

Abstract

The generalized Petersen graph <span class="math inline">\(G(n,k)\)</span> is a cubic graph with vertex set <span class="math inline">\(V(G(n,k))=\{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}\)</span> and edge set <span class="math inline">\(E(G(n,k))=\{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}\)</span> where the indices are taken modulo <span class="math inline">\(n\)</span>. Schwenk found the number of Hamiltonian cycles in <span class="math inline">\(G(n,2)\)</span>, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in <span class="math inline">\(G(n,3)\)</span> and <span class="math inline">\(G(n,4)\)</span>. This is attained by introducing <span class="math inline">\(G'(n,k)\)</span>, which is a modified version of <span class="math inline">\(G(n,k)\)</span>, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for <span class="math inline">\(k=3\)</span> and <span class="math inline">\(k=4\)</span>, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in <span class="math inline">\(G(n,k)\)</span> is a sum of the number of admissible subgraphs of <span class="math inline">\(G'(n,k)\)</span> over a certain subset of the classes.