Some Properties Related To Tiling "3.3.4.3.4"

Let D = [3.3.4.3.4] be an Archimedean tiling of the plane formed by regular tri-angles and squares, and let D denote the set of all the vertices of the tiling. A point ofD is called a D-point. In this thesis we mainly apply some methods used to discuss theproperties of the lattice points to investigate the properties related to D-points, and thenstudy the hamiltonian property of finite subgraphs of this Archimedean tiling D.Firstly we discuss the number of D-points lying on any given line in the plane, andprove that all the lines can be classified into five types according to the number of D-points lying on them, namely, no D-point, exactly one D-point, exactly two D-points,exactly four D-points and infinitely many D-points. We also characterize these five typesof lines.Secondly we define ST graph as the nontrivial, limited, two connected finite sub-graph of D such that the degree of any boundary vertex is less than or equal to four. Itis proved that except for three types of ST graphs, the other ST graphs are hamiltoniangraph.