Existence of 3-regular subgraphs in Cartesian product of cycles

Abstract

Let G be a graph obtained by taking the Cartesian product of finitely many cycles. It is known that G is bipancyclic, that is, G contains cycles of every even length from 4 to |V(G)|. We extend this result for the existence of 3-regular subgraphs in G. We prove that G contains a 3-regular, 2-connected subgraph with l vertices if l=8 or l=12 or l is an even integer with 16≤l≤|V(G)|. For l∈{6,10,14}, we give necessary and sufficient conditions for the existence of such subgraphs in G.