Title:

Thresholds and the structure of sparse random graphs

In this thesis, we obtain approximations to the non3colourability threshold of sparse random graphs and we investigate the structure of random graphs near the region where the transition from 3colourability to non3colourability seems to occur. It has been observed that, as for many other properties, the property of non3colourability of graphs exhibits a sharp threshold behaviour. It is conjectured that there exists a critical average degree such that when the average degree of a random graph is around this value the probability of the random graph being non3colourable changes rapidly from near 0 to near 1. The difficulty in calculating the critical value arises because the number of proper 3colourings of a random graph is not concentrated: there is a `jackpot' effect. In order to reduce this effect, we focus on a subfamily of proper 3colourings, which are called rigid 3colourings. We give precise estimates for their expected number and we deduce that when the average degree of a random graph is bigger than 5, then the graph is asymptotically almost surely not 3colourable. After that, we investigate the non$k$colourability of random regular graphs for any $k \geq 3$. Using a first moment argument, for each $k \geq 3$ we provide a bound so that whenever the degree of the random regular graph is bigger than this, then the random regular graph is asymptotically almost surely not $k$colourable. Moreover, in a (failed!) attempt to show that almost all 5regular graphs are not 3colourable, we analyse the expected number of rigid 3colourings of a random 5regular graph. Motivated by the fact that the transition from 3colourability to non3colourability occurs inside the subgraph of the random graph that is called the 3core, we investigate the structure of this subgraph after its appearance. Indeed, we do this for the $k$core, for any $k \geq 2$; and by extending existing techniques we obtain the asymptotic behaviour of the proportion of vertices of each fixed degree. Finally, we apply these results in order to obtain a more clear view of the structure of the 2core (or simply the core) of a random graph after the emergence of its giant component. We determine the asymptotic distributions of the numbers of isolated cycles in the core as well as of those cycles that are not isolated there having any fixed length. Then we focus on its giant component, and in particular we give the asymptotic distributions of the numbers of 2vertex and 2edgeconnected components.
