Some Extremal And Structural Problems In Graph Theory

This work considers three main topics. In Chapter 2, we deal with König-Egerváry graphs. We will give two new characterizations of König-Egerváry graphs as well as prove a related lower bound for the independence number of a graph. In Chapter 3, we study joint degree vectors (JDV). A problem arising from statistics is to determine the maximum number of non-zero elements of a JDV. We provide reasonable lower and upper bounds for this maximum number. Lastly, in Chapter 4 we study a problem in chemical graph theory. In particular, we characterize extremal cases for the number of maximal matchings in two linear polymers of chemical interest: the polyspiro chains and benzenoid chains. We also enumerate maximal matchings in several classes of these linear polymers and use the obtained results to determine the asymptotic behavior of these matchings

KE Theory & the Number of Vertices Belonging to All Maximum Independent Sets in a Graph

For a graph $G$, let $\alpha (G)$ be the cardinality of a maximum independent set, let $\mu (G)$ be the cardinality of a maximum matching and let $\xi (G)$ be the number of vertices belonging to all maximum independent sets. Boros, Golumbic and Levit showed that in connected graphs where the independence number $\alpha (G)$ is greater than the matching number $\mu (G)$, $\xi (G) \geq 1 + \alpha(G) - \mu (G)$. For any graph $G$, we will show there is a distinguished induced subgraph $G[X]$ such that, under weaker assumptions, $\xi (G) \geq 1 + \alpha (G[X]) - \mu (G[X])$. Furthermore $1 + \alpha (G[X]) - \mu (G[X]) \geq 1 + \alpha (G) - \mu (G)$ and the difference between these bounds can be arbitrarily large. Lastly some results toward a characterization of graphs with equal independence and matching numbers is given

Bounds for the independence number of a graph

The independence number of a graph is the maximum number of vertices from the vertex set of the graph such that no two vertices are adjacent. We systematically examine a collection of upper bounds for the independence number to determine graphs for which each upper bound is better than any other upper bound considered. A similar investigation follows for lower bounds. In several instances a graph cannot be found. We also include graphs for which no bound equals $\alpha$ and bounds which do not apply to general graphs

Sufficient degree conditions for graph embeddings

In this dissertation, we focus on the sufficient conditions to guarantee one graph being the subgraph of another. In Chapter 2, we discuss list packing, a modification of the idea of graph packing. This is fitting one graph in the complement of another graph. Sauer and Spencer showed a sufficient bound involving maximum degrees, and this was further explored by Kaul and Kostochka to characterize all extremal cases. Bollobas and Eldridge (and independently Sauer and Spencer) developed edge sum bounds to guarantee packing. In Chapter 2, we introduce the new idea of list packing and use it to prove stronger versions of many existing theorems. Namely, for two graphs, if the product of the maximum degrees is small or if the total number of edges is small, then the graphs pack. In Chapter 3, we discuss the problem of finding k vertex-disjoint cycles in a multigraph. This problem originated from a conjecture of Erdos and has led to many different results. Corradi and Hajnal looked at a minimum degree condition. Enomoto and Wang independently looked at a minimum degree-sum condition. More recently, Kierstead, Kostochka, and Yeager characterized the extremal cases to improve these bounds. In Chapter 3, we improve on the multigraph degree-sum result. We characterize all multigraphs that have simple Ore-degree at least 4k -3 , but do not contain k vertex-disjoint cycles. Moreover, we provide a polynomial time algorithm for deciding if a graph contains k vertex-disjoint cycles. Lastly, in Chapter 4, we consider the same problem but with chorded cycles. Finkel looked at the minimum degree condition while Chiba, Fujita, Gao, and Li addressed the degree-sum condition. More recently, Molla, Santana, and Yeager improved this degree-sum result, and in Chapter 4, we will improve on this further

Application of graph theory to resource distribution policy-based synthesis of industrial symbiosis networks.

Masters Degree. University of KwaZulu-Natal, Durban.Industrial symbiosis (IS) involves the repurposing of waste and by-product streams from one chemical industry as feedstock to another. Given the growing environmental and economic concerns, it has become increasingly difficult for industries not to participate in IS. This has encouraged much research into the field, with IS network design being an important optimisation problem in the research space. However, challenges are associated with the creation of IS networks, with transportation costs and resource distribution being key factors. Furthermore, solution strategies are usually complex and neglect the structural features of the network. A possible solution is the use of graph theory for IS network creation. It was hypothesized that structural features of an IS network can evaluate the effect of distribution policies on IS networks created by graph matching algorithms. The Simplex method (SM), Edmonds-Karp algorithm (FF), and the Hungarian method (HM) were adapted to model IS networks, with the intention to establish a ranking in the suitability in creating IS networks. The adaption rendered the algorithms applicable to feasible IS network discovery under different distribution policies. This graph-based approach allowed for the seamless extraction of the network features as graph metrics. Rigorous testing of the adapted algorithms’ performance using graph metrics was done by simulating numerous IS scenarios. It was found that HM identified connections that, on average, minimised transportation costs to the greatest extent. The HM created networks with the smallest travelling distance than those of SM and FF, showing a 9 % and 6.06 % lower value than SM and FF, respectively. Furthermore, HM-IS networks created more stable and fair networks, which was inferred from the graph metrics. To confirm the HM’s apparent superiority in IS network creation, a case study was simulated with the defined distribution policies being modelled from the matching features. Each distribution policy was quantified as a cost from which it was found that HM-IS networks had a 72.5 % and 74.9 % lower overall distribution cost than FF-IS networks and SM-IS networks, respectively. It was concluded that HM is the most suited for IS network creation and that graph-based modelling of IS is a feasible approach.Spelling error in title in original