Mutually independent Hamiltonianicity of Cartesian product graphs

Abstract

Two Hamiltonian cycles $C_1=\langle u_0,u_1,u_2,...,u_{n-1},u_0 \rangle$ and $C_2=\langle v_0,v_1,v_2,...,v_{n-1},v_0 \rangle$ of a graph G are independent starting at $u_0$ if $u_0=v_0, u_i\ne v_i$ for all $1\le i\le n-1$. A set of Hamiltonian cycles C of G are k-mutually independent starting at vertex u if any two different Hamiltonian cycles of C are independent starting at u and $|C| = k$. The mutually independent Hamiltonianicity of graph G is the maximum integer k, such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles starting at u, denoted by IHC($G)=k$. The Cartesian product of graphs G and H, written by $G \times H$, is the graph with vertex set $V(G) \times V(H)$ specified by putting (u, v) adjacent to $(u', v')$ if and only if $(1)\;u = u'$ and $vv' \in E(H),$ or $(2)\;v = v'$ and $uu' \in E(G)$. In this paper, for $G = G_1 \times G_2$, where $G_1$ and $G_2$ are Hamiltonian graphs, IHC($G_1 \times G_2) \ge$ IHC($G_1)$ or IHC($G_1)$ + 2 is proved when given some different conditions.