## Graduate Theses, Dissertations, and Problem Reports

#### Title

Cycles and Bases of Graphs and Matroids

Summer

2012

Dissertation

PhD

#### College

Eberly College of Arts and Sciences

Mathematics

Hong-Jian Lai

#### Abstract

The objective of this dissertation is to investigate the properties of cycles and bases in matroids and in graphs. In , Tutte defined the circuit graph of a matroid and proved that a matroid is connected if and only if its circuit graph is connected. Motivated by Tutte's result, we introduce the 2nd order circuit graph of a matroid, and prove that for any connected matroid M other than U1,1, the second order circuit graph of M has diameter at most 2 if and only if M does not have a restricted minor isomorphic to U2,6.;Another research conducted in this dissertation is related to the eulerian subgraph problem in graph theory. A graph G is eulerian if G is connected without vertices of odd degrees, and G is supereulerian if G has a spanning eulerian subgraph. In , Boesch, Suffey and Tindel raised a problem to determine when a graph is supereulerian, and they remarked that such a problem would be a difficult one. In , Pulleyblank confirmed the remark by showing that the problem to determine if a graph is supereulerian, even within planar graphs, is NP-complete. Catlin in  introduced a reduction method based on the theory of collapsible graphs to search for spanning eulerian subgraphs in a given graph G. In this dissertation, we introduce the supereulerian width of a graph G, which generalizes the concept of supereulerian graphs, and extends the supereulerian problem to the supereulerian width problem in graphs. Further, we also generalize the concept of collapsible graphs to s-collapsible graphs and develop the reduction method based on the theory of s-collapsible graphs. Our studies extend the collapsible graph theory of Catlin. These are applied to show for any integer n > 2, the complete graph Kn is (n - 3)- collapsible, and so the supereulerian width of Kn is n - 2. We also prove a best possible degree condition for a simple graph to have supereulerian width at least 3.;The number of edge-disjoint spanning trees plays an important role in the design of networks, as it is considered as a measure of the strength of the network. As disjoint spanning trees are disjoint bases in graphic matroids, it is important to study the properties related to the number of disjoint bases in matroids. In this dissertation, we develop a decomposition theory based on the density function of a matroid, and prove a decomposition theorem that partitions the ground set of a matroid M into subsets based on their densities. As applications of the decomposition theorem, we investigate problems related to the properties of disjoint bases in a matroid. We showed that for a given integer k > 0, any matroid M can be embedded into a matroid M' with the same rank (that is, r(M) = r( M')) such that M' has k disjoint bases. Further we determine the minimum value of |E( M')| -- |E(M)| in terms of invariants of M. For a matroid M with at least k disjoint bases, we characterize the set of elements in M such that removing any one of them would still result in a matroid with at least k disjoint bases.

COinS

#### DOI

https://doi.org/10.33915/etd.4886