Hamiltonicity and pancyclicity of cartesian products of graphs

Cycles in complementary prisms

“…Hence v 1 ̸ ∈ U (2) . This implies that F (2) = F and hence f (T ) = f (T (2) ) − 2. By induction, there are cycles C (2)− and C (2) in T (2) T (2) such that (T (2) , F (2) , U (2) , C (2)− , C (2) ) is good.…”

“…By induction, there are cycles C (2)− and C (2) in T (2) T (2) such that (T (2) , F (2) , U (2) , C (2)− , C (2) ) is good. Since v 1 has degree 2 in T (2) T (2) , the cycle C (2) contains the path v 1 v 1 v 2 . Since C (2) is very good, it contains three independent edges, say v a v b , v c v d , and v e v f , of T (2) .…”

“…Next we assume that v 2 ∈ U (1) . For a contradiction, we assume that v 1 ∈ U (2) . This implies that v 2 ∈ U (1) ∩ N (2) .…”

“…For a contradiction, we assume that v 1 ∈ U (2) . This implies that v 2 ∈ U (1) ∩ N (2) . By definition, every vertex in U (1) ∩ N (2) has at least 2 neighbors in T (1) that belong to U (2) ∩ N (1) and every vertex in U (2) ∩ N (1) has at least 2 neighbors in T (1) that belong to U (1) ∩ N (2) , that is, v 2 lies in a subgraph of T (1) of minimum degree at least 2, which is impossible because T is a tree.…”

“…A related open conjecture posed by Rosenfeld and Barnette [15] claims that the prism of a 3-connected planar graph is Hamiltonian. Hamiltonicity and pancyclism of prisms were studied for instance in [1,2,6,7,14], see [10] and Section 30.2 of [8] for further discussion of the background and partial results.…”