On fractional version of oriented coloring

TL;DR

This paper introduces and studies the fractional oriented coloring, revealing its properties for directed cycles and sparse planar graphs, and establishing bounds related to girth and chromatic number.

Contribution

It defines the fractional version of oriented coloring and provides foundational results, including bounds for planar graphs with large girth and exact values for directed cycles.

Findings

For large girth planar graphs, fractional oriented chromatic number is at most 4+ε.

Existence of oriented planar graphs with girth g_ε having chromatic number 5.

Exact fractional oriented chromatic number for directed cycles depends on prime divisors of cycle length.

Abstract

We introduce the fractional version of oriented coloring and initiate its study. We prove some basic results and study the parameter for directed cycles and sparse planar graphs. In particular, we show that for every $\epsilon > 0$, there exists an integer $g_{\epsilon} \geq 12$ such that any oriented planar graph having girth at least $g_{\epsilon}$ has fractional oriented chromatic number at most $4+\epsilon$. Whereas, it is known that there exists an oriented planar graph having girth at least $g_{\epsilon}$ with oriented chromatic number equal to $5$. We also study the fractional oriented chromatic number of directed cycles and provide its exact value. Interestingly, the result depends on the prime divisors of the length of the directed cycle.