余可图子拟阵中合格子集的存在性

Existence of eligible subsets in the cographic submatroid

根据Seymour分解定理,一个3-连通的正则拟阵如果不是可图的,余可图的,也不同构于二元域上的一个5行10列矩阵对应的向量拟阵R_(10),那么这个正则拟阵一定可以写成其中两个子式的3-和,而两个子式中有一个子式是可图的或者余可图的.特别地,当其中一个子式是余可图拟阵时,如果这个子式中存在非空合格子集,那么正则拟阵的超欧拉性与它收缩这个合格子集后所得子拟阵的超欧拉性等价.本文讨论了此类正则拟阵M在余围长不小于max{(r(M)+1)/10,8}且围长不小于4时非空合格子集的存在性.

According to Seymour’s decomposition theorem,when a 3-connected regular matroid is neither graphic,cographic,nor isomorphic to R_(10)(the vector matroid of a five rows ten columns matrix on binary field),it must be the 3-sum of its two minors,one of which is graphic or cographic.Moreover,if one of the minors is cographic and has a non-empty eligible subset,then supereulerian of the regular matroid is equivalent to supereulerian of its minors obtained by contracting the eligible subset.In this paper,we discuss the existence of the non-empty eligible subsets in this type of regular matroid M where its cogirth is no less than max{(r(M)+1)/10,8}and girth is no less than 4.