| A plane tiling ? is a countable family of closed sets ? ={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. |
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