1‐Factorizations of cartesian products of regular graphs

Abstract

AbstractLet G1, G2…, Gn be regular graphs and H be the Cartesian product of these graphs (H = G1 × G2 × … × Gn). The following will be proved: If the set {G1, G2…, Gn} has at leat one of the following properties: (*) for at leat one i ϵ {1, 2,…, n}, there exists a 1‐factorization of Gi or (**) there exists at least two numbers i and j such that 1 ≤ i < j ≤ n and both the Graphs Gi and Gj contain at least one 1‐factor, then there exists a 1‐factorization of H. Further results: Let F be a cycle of length greater than three and let G be an arbitrary cubic graph. Then there exists a 1‐factorization of the 5‐regular graph H = F × G. The last result shows that neither (*) nor (**) is a necessary condition for the existence of a 1‐factorization of a Cartesian product of regular graphs.