Reverse Mathematics and the Coloring Number of Graphs

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We use methods of reverse mathematics to analyze the proof theoretic strength of a theorem 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 for which below any vertex in V there are fewer than κ many vertices connected to it by E. We will study a theorem due to Komjáth and Milner, stating that if a graph is the union of n forests, then the coloring number of the graph is at most 2n. We focus on the case when n=1.