Reverse Mathematics and the Coloring Number of Graphs

Abstract

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 $\kappa$ such that there is a well-ordering of $V$ for which below any vertex in $V$ there are fewer than $\kappa$ 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 $n=1$.