Hamiltonicity and Eulerianity of Some Bipartite Graphs Associated to Finite Groups

Let G be a finite group. Associate a simple undirected graph Γ_G with G, called bipartite graph associated to elements and cosets of subgroups of G, as follows : Take G ∪ S_G as the vertices of Γ_G, with S_G is the set of all subgroups of a group G and join two vertices a ∈ G and H ∈ S_G if and only if aH = Ha. In this paper, hamiltonicity and eulerianity of Γ_G for some finite groups G are studied. In particular, it is obtained that for any cyclic group G, Γ_G is hamiltonian if and only if |G| = 2 and Γ_G is eulerian if and only if |G| is even non-perfect square number. Also, we prove that Γ_Dn is eulerian if k is even and n = 2k and for some other cases of n, Γ_Dn is not eulerian.