1
$\begingroup$

The Hajós number of a graph $G$ is the largest $k$ such that there are $k$ vertices in $G$ with a path between each pair so that all $k \choose 2$ paths are internally pairwise vertex disjoint (i.e. if a vertex is not one of the $k$, then it appears on at most one such path), and none of the $k$ vertices is an internal vertex of any of the paths. Is there a graph whose chromatic number exceeds twice its Hajos number?

The problem is from the book Probabilistic Methods by Noga Alon.

  • 0
    What does that book indicate about the status of this problem? Is it said to be an "open problem", or is it simply assigned as an exercise for the Reader? – 2017-02-03
  • 0
    See [The Probabilistic Method (3rd. ed.)](http://onlinelibrary.wiley.com/book/10.1002/9780470277331) by Noga Alon (Wiley, 2008). – 2017-02-03
  • 0
    @hardmath I have seen the forth edition. But it only has a hint to consider G(n,0.5) – 2017-02-03

1 Answers 1

0

Hint: Consider $G(n,\frac{1}{2})$

Every set contains almost half of edges $w.h.p.$

So you need at least one vertex for almost half on $k\choose2$ which is greater than total number of vertices if $k$ is large enough.