On fractional version of oriented coloring
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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 g_ϵ≥ 12 such that any oriented planar graph having girth at least g_ϵ has fractional oriented chromatic number at most 4+ϵ. Whereas, it is known that there exists an oriented planar graph having girth at least g_ϵ 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.
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