Study On The Supereulerian Properties Of Connected Regular Matroids

For a nonempty finite set E called ground set,if I is a subset of 2E,and satisfies the following so-called independent sets axioms:(I1)(?)∈I;(I2)if I ∈ I,and I’(?)I,then,I’∈I;(I3)if I1,I2 ∈ I and |I1|<|I2|,then there exists e ∈I2-I1 such that I1∪e ∈ I.Then the ordered pair M=(E,I)is called a matroid on the ground set E,each element contained in I is an independent set and otherwise a dependent set of the matroid.A matroid is called a regular matroid if it is isomorphic to the vector space of a total unitary matrix over a binary field.A minimal dependent set is called a circuit.Through the thesis,C(M)represents a set consists of all minimal dependent sets of M.If any two elements of E are in the same circuit,then the matroid is called connected(or 2-connected matroid).The maximal independent set in a matroid is called a base,and B(M)is the set of all bases in a matroid.According to the base axioms which is equivalent with the independent set axioms,any two bases have same cardinality called the rank of matroid and denoted by r(M).Generally,the cycle C in a matroid is a disjoint union of the minimal dependent sets.If there is a cycle C in a matroid M with r(C)=r(M),C is called the spanning cycle of M.And such matroid M is called supereulerian.The connected even graph is called Eulerian,and the graph containing a spanning Euler subgraph is defined as supereulerian graph.Therefore,supereulerian problem of matroid is an development of corresponding problems of graph.This thesis focuses on the supereulerian property of connected regular matroids.Based on the work of former people,the thesis discusses that what conditions should a regular matroid must have so that it can be supereulerian.Following results are proved:(1)Let M be a connected simple regular matroid.If every cocircuit D of M satisfies |D|>max{(r(M)+1)/10,9},then M is supereulerian.(2)For any real number c(0<c<1)there exists an integer f(c)such that if every cocircuit D of a connected simple cographic matroid M satisfies |D|>max{c(r(M))+1,f(c)},then M is supereulerian.Based on Seymour Decomposition theorem of regular matroids,the problem is reduced to the cases of graphic minor and cographic minor by using minimal counter-examples and mathematical induction.Finally,by utilizing the approaches of strength and fractional arboricity of a matroid,it is determined that a lower bound of cogirth in a regular matroid having spanning cycles,which is relevant with the rank of the matroid.A affirmative conclusion is obtained for the possibility of lower bound in the case of general constant.