A Note On K-ordered Hamiltonian Graphs And The Complete Arcs In Tiny Projective Plane

One of the focuses in the graph theory is the Hamiltonian problems, but haven't been solved completely. Cayley graphs is the graphs over the group, and all of the Cayley graphs over the Able group are Hamiltonian graphs. The additive group of the integers modulo n is an Able group, And( k ) is the Cayley graph over the additive group of the integers modulo 3( k -1), consequently, it is a Hamiltonian graph.The definition of k -ordered Hamiltonian graph was given by Lenhard in 1997, but the Cayley graphs'k -ordered Hamiltonicity haven't been given at yet. As And(3) is not 4-ordered, this paper concerns the 4-ordered Hamiltonicity of And( k ) while k≥4. We find an useful classify and get the fundamental results: And( k ) is 4-ordered Hamiltonian graph.Finally, we study some of the complete arcs in tiny projective plane. The k - arcsover projective plane was raised by Segre in 1955, and the problem that find the maximum k - arcsin PG( r , q ) haven't been solved. We concern the incidence graphs of the projective plane, some of the complete arcs in tiny projective plane are obtained while r =2, which is another main result of this paper.