Proper orientations and proper chromatic number

TL;DR

This paper proves bounds on the proper chromatic number of graphs, including planar graphs, and introduces fractional orientations as a novel tool for analyzing graph colorings and orientations.

Contribution

It resolves two major conjectures about the proper chromatic number, establishing bounds for planar graphs and general graphs using fractional orientations.

Findings

Proper chromatic number of planar graphs is at most 14.

Proper chromatic number is bounded by a function of chromatic number and maximum average degree.

Introduces fractional orientations as a new analytical tool.

Abstract

The proper chromatic number $\Vec{\chi}(G)$ of a graph $G$ is the minimum $k$ such that there exists an orientation of the edges of $G$ with all vertex-outdegrees at most $k$ and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper chromatic number are resolved. First it is shown, that $\Vec{\chi}(G)$ of any planar graph $G$ is bounded (in fact, it is at most 14). Secondly, it is shown that for every graph, $\Vec{\chi}(G)$ is at most $O(\frac{r\log r}{\log\log r})+\tfrac{1}{2}\MAD(G)$, where $r=\chi(G)$ is the usual chromatic number of the graph, and $\MAD(G)$ is the maximum average degree taken over all subgraphs of $G$. Several other related results are derived. Our proofs are based on a novel notion of fractional orientations.