## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 57, Number 1 (2016), 27-44.

### 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 $n=1$.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 57, Number 1 (2016), 27-44.

**Dates**

Received: 26 October 2011

Accepted: 9 August 2013

First available in Project Euclid: 30 September 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1443619666

**Digital Object Identifier**

doi:10.1215/00294527-3321905

**Mathematical Reviews number (MathSciNet)**

MR3447723

**Zentralblatt MATH identifier**

1353.03008

**Subjects**

Primary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03D80: Applications of computability and recursion theory

Secondary: 05C15: Coloring of graphs and hypergraphs

**Keywords**

reverse mathematics coloring number graph computability theory

#### Citation

Jura, Matthew. Reverse Mathematics and the Coloring Number of Graphs. Notre Dame J. Formal Logic 57 (2016), no. 1, 27--44. doi:10.1215/00294527-3321905. https://projecteuclid.org/euclid.ndjfl/1443619666