### Shai Gutner

Tel-Aviv University, 1992

## Choice numbers of graphs

### Abstract

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not
necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same authors, is settled,
proving that every $2$-choosable graph is also
$(4:2)$-choosable. Applying probabilistic methods, an upper bound
for the $k^{th}$ choice number of a graph is given. We also prove that a
directed graph with maximum outdegree $d$ and no odd directed
cycle is $(k(d+1):k)$-choosable for every $k \geq 1$.
Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number).
We conclude with complexity analysis of some decision problems related to graph choosability.

### Table of Contents

1. Introduction
2. A solution to a problem of Erd\H{o}s, Rubin and Taylor
3. An upper bound for the $k$th choice number
4. Choice numbers and orientations
5. Properties of $(2k:k)$-choosable graphs
6. The complexity of graph choosability
7. The strong choice number