# Fractional cocoloring of graphs

## TL;DR

This paper introduces and analyzes the fractional cochromatic number of graphs, establishing bounds and relationships with fractional chromatic number, and exploring extremal values across different graph classes and surfaces.

## Contribution

It extends the theory of graph coloring by defining fractional cochromatic number and proves key bounds and properties paralleling classical results.

## Key findings

- For triangle-free graphs, fractional cochromatic number equals fractional chromatic number except for stars.
- If a graph contains no k-clique, then Z_f(G) ≤ χ_f(G) ≤ Z_f(G) + R(k,k).
- Maximum fractional cochromatic number over n-vertex graphs is Θ(n / log n).

## Abstract

The cochromatic number $Z(G)$ of a graph $G$ is the fewest number of colors needed to color the vertices of $G$ so that each color class is a clique or an independent set. In a fractional cocoloring of $G$ a non-negative weight is assigned to each clique and independent set so that for each vertex $v$, the sum of the weights of all cliques and independent sets containing $v$ is at least one. The smallest total weight of such a fractional cocoloring of $G$ is the fractional cochromatic number $Z_f(G)$. In this paper we prove results for the fractional cochromatic number $Z_f(G)$ that parallel results for $Z(G)$ and the well studied fractional chromatic number $\chi_f{(G)}$. For example $Z_f(G)=\chi_f(G)$ when $G$ is triangle-free, except when the only nontrivial component of $G$ is a star. More generally, if $G$ contains no $k$-clique, then $Z_f(G)\le \chi_f(G)\le Z_f(G)+R(k,k)$. Moreover, every graph $G$ with $\chi_f(G)=m$ contains a subgraph $H$ with $Z_f(H)\ge (\frac 14 - o(1))\frac m{\log_2 m}$. We also prove that the maximum value of $Z_f(G)$ over all graphs $G$ of order $n$ is $\Theta (n/\log n)$, and the maximum over all graphs embedded on an orientable surface of genus $g$ is $\Theta(\sqrt g / \log g)$.