Study On Matroids' Circuit Graphs And Supereulerian Property Of Dual Matroids

The concepts of matroid is the abstraction of of linear algebra and graph theory,so it widely borrows relevant terms and abstracts the lots of properties of graph.The circuit graphs of a matroid is a new concept to characterize the circuit structure of matroids.Tutte put forward a first definition of matroids' circuit graph in 1965 and proved that matroids are connected if and only if the circuit graph of matroids is connected.Different from Tutte's definition,Li defined another kind of circuit graph of matroid,and obtained some good conclusions for the first-order circuit graph,including a lower bound of the connectivity and the uniform Hamiltonian of a connected matroid first-order circuit graph.The second-order circuit graph of matroid has more complex structural requirements than the first-order circuit graph.So far,there are few relevant research results.We have conducted a preliminary study on this problem and obtained some results.In addition,the research on supereulerian of matroid and its dual are also a blank.In the case of graphs,Professor Lai and others have studied the supereulerian of graph and its complement.Inspired by their researches,we preliminarily study the supereulerian of a matroid and its dual,and obtain some results.In this thesis,we mainly study the Hamiltonian connectivity of general matroid first-order circuit graph,the connectivity and Hamiltonian of second-order circuit graph of uniform matroids,and the supereulerian of binary matroids and their dual.(1)Based on the research results of Li and others on matroid circuit graphs and the conclusions of Oxley and others on the connectivity and circuit structure of matroids,the Hamiltonian connectivity of first-order circuit graphs corresponding to matroids is studied,and the Hamiltonian connectivity of first-order circuit graphs of general matroids is proved;(2)By using the structural features of uniform matroids and the method of contracting an element,the first-order circuit graph and second-order circuit graph of a uniform matroid are skillfully connected.So that,we proved the connectivity and Hamiltonian of the second-order circuit graph of uniform matroids.(3)By using the relationship between the edge arboricity of a planar graph and the maximum number of edge disjoint spanning trees in a dual plane graph,some sufficient conditions for the dual plane graphs to be supereulerian are obtained.Simultaneously,for the supereulerian of dual matroids of binary matroids,some sufficient conditions related to the maximum number of disjoint bases is given.