HAMILTONICITY AND EULERIANITY OF SOME BIPARTITE GRAPHS ASSOCIATED TO FINITE GROUPS

Let G be a finite group. Associate a simple undirected graph ΓG with G, called a bipartite graph associated to elements and cosets of subgroups of G, as follows : Take G ∪ SG as the vertices of ΓG, with SG is the set of all subgroups of group G and a ∈ G and H ∈ SG if and only if aH = Ha. In this paper, hamiltonicity and Eulerianity of ΓG for some finite groups G are studied. In particular, the results
obtained that for any cyclic group G, ΓG is hamiltonian if and only if |G| = 2 and ΓG is Eulerian if and only if |G| is an even non-perfect square number. Also, we prove that ΓDn
is Eulerian if k is even and n = 2k and Γ(Dn) is not Eulerian for some other cases of n.