On fractional version of oriented coloring

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 ε>0, there exists an integer gε≥12 such that any oriented planar graph having girth at least gε has fractional oriented chromatic number at most 4+ε. Whereas, it is known that there exists an oriented planar graph having girth at least gε 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.