Reverse Mathematics and the coloring number of graphs

Date of Completion

January 2009






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 2n. 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. ^