Infinite graphs, graph-like spaces and B-matroids

The central theme of this thesis is to prove results about infinite mathematical objects by studying the behaviour of their finite substructures. In particular, we study B-matroids, which are an infinite generalization of matroids introduced by Higgs \cite{higgs}, and graph-like spaces, which are topological spaces resembling graphs, introduced by Thomassen and Vella \cite{thomassenvella}. Recall that the circuit matroid of a finite graph is a matroid defined on the edges of the graph, with a set of edges being independent if it contains no circuit. It turns out that graph-like continua and infinite graphs both have circuit B-matroids. The first main result of this thesis is a generalization of Whitney's Theorem that a graph has an abstract dual if and only if it is planar. We show that an infinite graph has an abstract dual (which is a graph-like continuum) if and only if it is planar, and also that a graph-like continuum has an abstract dual (which is an infinite graph) if and only if it is planar. This generalizes theorems of Thomassen (\cite{thomassendual}) and Bruhn and Diestel (\cite{bruhndiestel}). The difficult part of the proof is extending Tutte's characterization of graphic matroids (\cite{tutte2}) to finitary or co-finitary B-matroids. In order to prove this characterization, we introduce a technique for obtaining these B-matroids as the limit of a sequence of finite minors. In \cite{tutte}, Tutte proved important theorems about the peripheral (induced and non-separating) circuits of a $3$-connected graph. He showed that for any two edges of a $3$-connected graph there is a peripheral circuit containing one but not the other, and that the peripheral circuits of a $3$-connected graph generate its cycle space. These theorems were generalized to $3$-connected binary matroids by Bixby and Cunningham (\cite{bixbycunningham}). We generalize both of these theorems to $3$-connected binary co-finitary B-matroids. Richter, Rooney and Thomassen \cite{richterrooneythomassen} showed that a locally connected, compact metric space has an embedding in the sphere unless it contains a subspace homeomorphic to $K_5$ or $K_{3,3}$, or one of a small number of other obstructions. We are able to extend this result to an arbitrary surface $\Sigma$; a locally connected, compact metric space embeds in $\Sigma$ unless it contains a subspace homeomorphic to a finite graph which does not embed in $\Sigma$, or one of a small number of other obstructions

Locally finite graphs with ends: A topological approach, I. Basic theory

AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested

Graphs, matroids, and geometric lattices

AbstractIt is shown that two triply connected graphs are isomorphic if their associated geometric lattices are isomorphic. The notion of vertex in a graph is described in terms of irreducible hyperplanes. Finally, necessary and sufficient conditions are given that a lattice be isomorphic to the geometric lattice associated with a graph

The Linkage Problem for Group-labelled Graphs

This thesis aims to extend some of the results of the Graph Minors Project of Robertson and Seymour to "group-labelled graphs". Let $\Gamma$ be a group. A $\Gamma$-labelled graph is an oriented graph with its edges labelled from $\Gamma$, and is thus a generalization of a signed graph. Our primary result is a generalization of the main result from Graph Minors XIII. For any finite abelian group $\Gamma$, and any fixed $\Gamma$-labelled graph $H$, we present a polynomial-time algorithm that determines if an input $\Gamma$-labelled graph $G$ has an $H$-minor. The correctness of our algorithm relies on much of the machinery developed throughout the graph minors papers. We therefore hope it can serve as a reasonable introduction to the subject. Remarkably, Robertson and Seymour also prove that for any sequence $G_1, G_2, \dots$ of graphs, there exist indices $i<j$ such that $G_i$ is isomorphic to a minor of $G_j$. Geelen, Gerards and Whittle recently announced a proof of the analogous result for $\Gamma$-labelled graphs, for $\Gamma$ finite abelian. Together with the main result of this thesis, this implies that membership in any minor closed class of $\Gamma$-labelled graphs can be decided in polynomial-time. This also has some implications for well-quasi-ordering certain classes of matroids, which we discuss

Extensions of Signed Graphs

Given a signed graph (G, Σ) with an embedding on a surface S, we are interested in "extending" (G, Σ) by adding edges and splitting vertices, such that the resulting graph has no embedding on S. We show (assuming 3-connectivity for (G, Σ)) that there are a small number of minimal extensions of (G, Σ) with no such embedding, and describe them explicitly. We also give conditions, for several surfaces S, for an embedding of a signed graph on S to extend uniquely. These results find application in characterizing the signed graphs with no odd-K_5 minor

Inequivalent Representations of Matroids over Prime Fields

It is proved that for each prime field $GF(p)$, there is an integer $f(p)$ such that a 4-connected matroid has at most $f(p)$ inequivalent representations over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term "$k$-coherence". We obtain a variety of other results on inequivalent representations including the following curious one. For a prime power $q$, let ${\mathcal R}(q)$ denote the set of matroids representable over all fields with at least $q$ elements. Then there are infinitely many Mersenne primes if and only if, for each prime power $q$, there is an integer $m_q$ such that a 3-connected member of ${\mathcal R}(q)$ has at most $m_q$ inequivalent GF(7)-representations. The theorems on inequivalent representations of matroids are consequences of structural results that do not rely on representability. The bulk of this paper is devoted to proving such results

Graph theory in America 1876-1950

This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory