| A plane tiling Τ is a countable family of closed sets Τ ={Ti:i ∈ I},I={1,2,…},which satisfies ∪Ti =R2,and int Ti ∩ int Tj =（?）（i ≠ j）.If the corners and sides of a poly-gon coincide with the vertices and edges of the tiling,then we say the tiling by polygons is edge-to-edge.Archimedean tilings are edge-to-edge tilings by regular polygons,satisfy-ing all vertices are of the same type.There are exactly 11 distinct Archimedean tilings denoted by（36）,（44）,（63）,（3.122）,（33.42）,（34.6）,（3.6.3.6）,（32.4.3.4）,（4.82）,（3.4.6.4）and（4.6.12）.An edge-to-edge tiling by regular polygons is called 2-uniform if its vertices form pre-cisely 2 transitivity classes with respect to the group of symmetries of the tiling.There exist 20 distinct types of 2-uniform edge-to-edge tilings by regular polygons,namely:（36;33.42）1,（36;33.42）2,（36;34.6）1,（36;34.6）2,（33.42;44）1,（33.42;44）2,（33.42;32.4.3.4）1,（33.42;32.4.3.4）2,（33.42;3.4.6.4）,（36;32.4.3.4）,（32.4.3.4;3.4.6.4）,（34.6;32.62）,（36;32.62）,（32.62;3.6.3.6）,（3.42.6;3.6.3.6）1,（3.42.6;3.6.3.6）2,（3.42.6;3.4.6.4）,（36;32.4.12）,（3.4.6.4;4.6.12）and（3.4.3.12;3.122）.Gallai’s property means that in a connected graph,all longest paths or cycles have empty intersection.We focus on the property about cycles in the thesis.In Chapter 2,we study Gallai’s property in Archimedean tiling graphs and 2-uniform tiling graphs.Basing on Lemma 2.1 and concrete constructions,we prove that there exist 2-connected subgraphs in Archimedean tiling graphs（34.6）,（33.42）,（32.4.3.4）,（3.6.3.6）,（3.4.6.4）,（4.82）,（4.6.12）,（3.122）and all 2-uniform tiling graphs（36;33.42）1,（36;33.42）2,（36;34.6）1,（36;34.6）2,（33.42;44）1,（33.42;44）2,（33.42;32.4.3.4）1,（33.42;32.4.3.4）2,（33.42;3.4.6.4）,（36;32.4.3.4）,（32.4.3.4;3.4.6.4）,（34.6;32.62）,（36;32.62）,（32.62;3.6.3.6）,（3.42.6;3.6.3.6）1,（3.42.6;3.6.3.6）2,（3.42.6;3.4.6.4）,（36;32.4.12）,（3.4.6.4;4.6.12）,（3.4.3.12;3.122）,with order 52,35,53,65,58,98,130,188,33,33,33,35,35,35,35,35,35,40,49,63,65,65,70,65,78,105,123,188 respectively,satisfying Gallai’s property.In Chapter 3,we prove the existence of 3-connected subgraphs with Gallai’s prop-erty in Archimedean tilings（36）and（33.42）,where the method entirely differs from the previous proofs for graphs satisfying Gallai’s property. |
PDF Full Text Request