A proof of Menger's theorem by contraction

A short proof of the classical theorem of Menger concerning the number of disjoint AB-paths of a finite graph for two subsets A and B of its vertex set is given. The main idea of the proof is to contract an edge of the graph

Branch-width and well-quasi-ordering in matroids and graphs

AbstractWe prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off

Eulerian subgraphs containing given vertices and hamiltonian line graphs

AbstractLet G be a graph and let D1(G) be the set of vertices of degree 1 in G. Veldman (1994) proves the following conjecture from Benhocine et al. (1986) that if G − D1(G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E(G), d(x) + d(y) > (2n)/5 − 2, then for n large, L(G), the line graph of G, is hamiltonian. We shall show the following improvement of this theorem: if G − D1(G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E(G), max[;d(x), d(y)] ⩾ n/5 − 1, then for n large, L(G) is hamiltonian with the exception of a class of well characterized graphs. Our result implies Veldman's theorem

群ラベル付きグラフにおける組合せ最適化

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 岩田 覚, 東京大学教授 定兼 邦彦, 東京大学教授 今井 浩, 国立情報学研究所教授 河原林 健一, 東京大学准教授 平井 広志University of Tokyo(東京大学

Well-Quasi-Ordering by the Induced-Minor Relation

Robertson and Seymour proved Wagner\u27s Conjecture, which says that finite graphs are well-quasi-ordered by the minor relation. Their work motivates the question as to whether any class of graphs is well-quasi-ordered by other containment relations. This dissertation is concerned with a special graph containment relation, the induced-minor relation. This dissertation begins with a brief introduction to various graph containment relations and their connections with well-quasi-ordering. In the first chapter, we discuss the results about well-quasi-ordering by graph containment relations and the main problems of this dissertation. The graph theory terminology and preliminary results that will be used are presented in the next chapter. The class of graphs that is considered in this research is the class W of graphs that contain neither W4 (a wheel graph with five vertices) and K5\e (a complete graph on five vertices minus an edge) as an induced minor. Chapter 3 is devoted to studying the structure of this class of graphs. A class of graphs is well-quasi-ordered by a containment relation if it contains no infinite antichain, so infinite antichains are important. We construct in Chapter 4 an infinite antichain of W with respect to the induced minor relation and study its important properties in Chapter 5. These properties are used in determining all well-quasi-ordered subclasses of W to reach the main result of Chapter 6

A minor-monotone graph parameter based on oriented matroids

AbstractFor an undirected graph G = (V,E) let λ ′(G) be the largest d for which there exists an oriented matroid M on V of corank d such that for each nonzero vector (x+,x−) of M, x+ is nonempty and induces a connected subgraph of G.We show that λ′(G) is monotone under taking minors and clique sums. Moreover, we show that λ′(G) ⩽ 3 if and only if G has no K5- or V8-minor; that is, if and only if G arises from planar graphs by taking clique sums and subgraphs