# A Note on the Non-Colorability Threshold of a Random Graph

### Abstract

In this paper we consider the problem of establishing a value $r_0$ such that almost all random graphs with $n$ vertices and $rn$ edges, $r > r_0$, are asymptotically not 3-colorable. In our approach we combine the concept of rigid legal colorings introduced by Achlioptas and Molloy with the occupancy problem for random allocations of balls into bins. Using the sharp estimates obtained by Kamath et al. of the probability that no bin is empty after the random placement of the balls and exploiting the relationship between the placement of balls and the rigid legal colorings, we improve the value $r_0 = 2.522$ previously obtained by Achlioptas and Molloy to $r_0 = 2.495$.