Proper orientations and proper chromatic number

Abstract

The proper orientation number χ→(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 orientation number are resolved. First it is shown, that χ→(G) of any planar graph G is at most 14. Secondly, it is shown that for every graph, χ→(G) is at most ▪, where r=χ(G) is the usual chromatic number of the graph, and ▪ 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.