# What numbers are possible for the fractional chromatic numbers?

Question: What numbers are possible as fractional chromatic numbers?

Clearly, all nonnegative integers are possible, just use the complete graph $$K_n$$ on $$n$$ vertices. Also, if we limit our scope to only finite simple graphs, then irrational numbers cannot be fractional chromatic numbers.

For nonintegers, it is clear that fractional chromatic numbers have to be $$\geq 2$$, because if the graph has an edge, then at least $$2n$$ colours are needed for the two connected vertices if we assign $$n$$ colours to a vertex.

There are hence two main questions here:

1. Can every rational number $$\geq 2$$ be the fractional chromatic number of a finite simple graph?
2. Can every number $$\geq 2$$ be the fractional chromatic number of an infinite simple graph?
• The Kneser graph $KG(n, k)$ with $k<n-1$ has fractional chromatic number $n/k$, so that gives you all fractions $(2k+b)/k$ with $n=2k+b$ and $k\geq 2$. mathworld.wolfram.com/FractionalChromaticNumber.html Commented Nov 3, 2021 at 17:00

For 2, for irrational $$r>2$$, consider a strictly increasing sequence of rationals $$(r_i)_{i=1}^n$$ converging to $$r$$, where all $$r_i\geq2$$. Then take a disjoint union of all the Kneser graphs corresponding to each $$r_i$$. That is, given $$r_i=\frac{p_i}{q_i}$$, construct the Kneser graph $$KG(p_i, q_i)$$, and take the disjoint union of all these Kneser graphs.