Even-cycle decompositions of graphs with no odd-K4K_4-minor

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio

Characterization of Some Properties of Ribbon Graphs and Their Partial Duals

带子图可被看作是一个具有图结构的有边界的曲面，是胞腔嵌入图的一种表示形式.部分对偶推广了数学基本概念----胞腔嵌入图的几何对偶，它通过纽结的Jones多项式与图的Tutte型多项式之间建立关系，将纽结理论中各种版本的Thistlethwaite定理统一起来.部分对偶不但是几何对偶的深远扩展，而且在图论，拓扑学和物理学中有重要的应用. 本文刻画带子图及其部分对偶的欧拉和偶面图等若干性质.全文共分五章： 第一章首先概述本学位论文所研究问题的相关背景、国内外研究现状以及预备知识，然后简单介绍本文的主要结果、主要研究方案及结构安排. 第二章首先给出带子图严格的定义和例子，并且说明带子图与胞腔嵌...A ribbon graph is a surface with boundary and a cellularly embedded graph can be realized as a ribbon graph. The concept of partial dual generalizes the fundamental concept of the geometric dual of a cellularly embedded graph. It was introduced to unify various versions of Thistlethwaite theorems in knot theory that relate the Jones polynomial of knots with a Tutte-like polynomial of graphs. Parti...学位：理学博士院系专业：数学科学学院_应用数学学号：1902013015417

Centrally symmetric configurations of order polytopes

It is shown that the toric ideal of the centrally symmetric configuration of the order polytope of a finite partially ordered set possesses a squarefree quadratic initial ideal. It then follows that the convex polytope arising from the centrally symmetric configuration of an order polytope is a normal Gorenstein Fano polytope.Comment: 9 pages, Proof of Theorem 2.2 is simplified. Major revision on Section

Even Orientations and Pfaffian graphs

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments