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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 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.
