#### Title

Reverse Mathematics and the coloring number of graphs

#### Date of Completion

January 2009

#### Keywords

Mathematics

#### Degree

Ph.D.

#### Abstract

We use methods of Reverse Mathematics to analyze the proof theoretic strength of certain graph theoretic theorems involving the notion of coloring number. Classically, the coloring number of a graph *G* = (* V, E*) is the least cardinal κ such that there is a well ordering of *V* such that below any vertex in *V*, there are fewer than κ many vertices connected to it by *E*. A theorem which we will study in depth, due to Komjáth and Milner, states that if a graph is the union of *n* forests, then the coloring number of the graph is at most 2*n*. In particular, we look at the case when *n* = 1. In doing the above, it is necessary for us to formulate various different Reverse Mathematics definitions of coloring number; we also analyze the relationships between these definitions. ^

#### Recommended Citation

Jura, Matthew A, "Reverse Mathematics and the coloring number of graphs" (2009). *Doctoral Dissertations*. AAI3361011.

https://opencommons.uconn.edu/dissertations/AAI3361011