Fractional Coloring Methods with Applications to Degenerate Graphs and Graphs on Surfaces

Abstract

We study methods for finding strict upper bounds on the fractional chromatic number $\chi_f(G)$ of a graph $G$. We illustrate these methods by providing short proofs of known inequalities in connection with Grötzsch's 3-color theorem and the 5-color theorem for planar graphs. We also apply it to $d$-degenerate graphs and conclude that every $K_{d+1}$-free $d$-degenerate graph with $n$ vertices has independence number $<n/(d+1)$. We show that for each surface $S$ and every $\varepsilon>0$, the fractional chromatic number of any graph embedded on $S$ of sufficiently large width (depending only on $S$ and $\varepsilon$) is at most $4+\varepsilon$. In the same spirit we prove that Eulerian triangulations or triangle-free graphs of large width have $\chi_f\le 3+\varepsilon$, and quadrangulations of large width have $\chi_f\le 2+\varepsilon$. While the $\varepsilon$ is needed in the latter two results, we conjecture that in the first result $4+\varepsilon$ can be replaced by 4. The upper bounds $\chi_f\le 4+\varepsilon$, $\chi_f\le 3+\varepsilon$, $\chi_f\le 2+\varepsilon$, respectively, are already known for graphs on orientable surfaces, but our results are also valid for graphs on nonorientable surfaces. Surprisingly, a strict lower bound on the fractional chromatic number may imply an upper bound on the chromatic number: Grötzsch's theorem implies that every 4-chromatic planar graph $G$ has fractional chromatic number $\chi_f(G)\ge 3$. We conjecture that this inequality is always strict and observe that this implies the 4-color theorem for planar graphs.