Allan Bickle

Doctoral Dissertation Announcement

---

Candidate: Allan Bickle

Degree of: Doctor of Philosophy

Department: Mathematics

Title: The K-cores of a Graph

Committee:
Dr. Allen Schwenk, Chair
Dr. Ping Zhang
Dr. Arthur White
Dr. Jeffrey Strom
Dr. Jonathan Hodge

Date: Thursday, October 28, 2010 2:00 p.m. - 4:00 p.m.
6625 Everett Tower

Abstract:
The k-core of a graph is the unique maximal subgraph of minimum degree at least k. The cores of a graph are nested, so their differences can be used to decompose a graph into k-shells.
Cores are useful for bringing some order to graphs without clear structure, including those arising from real-world applications. On the other hand, the k-shell decomposition leads naturally to the concept of monocore graphs, each of whose cores either are the entire graph or do not exist. Many familiar classes of graphs, such as regular graphs, trees, and complete multipartite graphs are monocore. Their structures can be investigated, along with the extremal monocore graphs, which are minimal or maximal with respect to being monocore. These are respectively k-collapsible graphs and maximal k-degenerate graphs. Cores have interesting interactions with graph operations including Cartesian products, joins, and line graphs.
Cores have a variety of applications to other areas of graph theory. They lead to natural and relatively efficient algorithms for many different varieties of graph coloring. They also have applications to the arboricity, domination numbers, planarity, Nordhaus-Gaddum theorems, and the Reconstruction Conjecture. They lead to an exact formula for a new variety of Ramsey numbers. In many applications, the 2-core is particularly important, as the 1-shell is composed of trees, which have been studied extensively. Results relating to these problems are presented.