Threshold phenomena in random graph colouring and satisfiability

We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph ' G'('n, p') on 'n' vertices formed by including each of the <math> <f> <fen lp="par"><stk><lyr>n</lyr><lyr>2</lyr></stk><rp post="par"></fen> </f> </math> possible edges independently of all others with probability ' p'. For a fixed integer 'k', let 'fk'(' n, d') = Pr['G'('n, d'/'n') is 'k'-colourable]. Erdös asked the following fundamental question: for 'k' >= 3, is there a constant 'c k' such that for any <math> <f> <g>e</g>>0,</f> </math> <display-math> <fd> <fl><lim align="c"><op><rf>lim</rf></op><ll>n[right arrow][infinity]</ll> </lim>f<inf>k</inf><fen lp="par">n,c<inf>k</inf>-<g>e</g><rp post="par"></fen> =1,<hsp sp="0.265"><hsp sp="0.265"><hsp sp="0.265"><rm>and</rm> <hsp sp="0.265"><hsp sp="0.265"><hsp sp="0.265"><lim align="c"><op><rf>lim</rf></op> <ll>n[right arrow][infinity]</ll></lim>f<inf>k</inf><fen lp="par">n,c<inf> k</inf>+<g>e</g><rp post="par"></fen>=0<hsp sp="2.000"><fen lp="par"> 1<rp post="par"></fen><hsp sp="0.265"><hsp sp="0.265"><hsp sp="0.265"> <hsp sp="0.265"><hsp sp="0.265"></fl> </fd> </display-math> We prove that for every 'k' >= 3, there exists a 'function' 'tk'('n') such that (1) holds upon replacing 'ck' by ' tk'('n'), thus establishing that indeed 'k'-colourability has a sharp threshold. Let <math> <f> d<inf>k</inf>=<rm>sup<fen lp="cub"><it>d&mid;<rm>lim<inf>n[right arrow] [infinity]</inf></rm>f<inf>k</inf></it><fen lp="par">n,d<rp post="par"></fen> =1<rp post="cub"></fen>.</rm></f> </math> Note that if 'ck' exists then, by definition, <math> <f> c<inf>k</inf>=d<inf>k</inf>.</f> </math> For the basic and most studied case 'k' = 3 we prove <display-math> <fd> <fl>3.84<d<inf>3</inf><5.05.</fl> </fd> </display-math> These are the best known bounds for 'd' 3. Our lower bound improves upon <math> <f> d<inf>3</inf>>=3.35&ldots;</f> </math> which corresponds to the critical probability for the emergence of a subgraph with minimum degree 3 in 'G'('n, p'). Thus, we also settle completely a long-standing question of Bollobás, establishing a (large) gap between the appearance of a subgraph with minimum degree 3 and of a 4-chromatic subgraph. The proof is algorithmic and we outline an asymptotically tight analysis of the underlying algorithm. Our upper bound improves upon 'd'3 < 5.15 of Molloy and Reed, illustrating how an idea of Kirousis et al., developed for random ' k'-SAT, can be viewed as a general refinement of the "first moment" method. Finally, we give the first mathematically rigorous results for a new model of random satisfiability which interpolates between random 2-SAT and random 3-SAT. The techniques are very similar with those developed for random graph colouring. Among other things we prove the following, rather surprising, proposition. It is well-known that a random formula on 'n' variables with 'rn' 2-clauses is almost surely (a.s.) satisfiable if 'r' < 1 and a.s. unsatisfiable if 'r' > 1. We prove that for any 'r' < 1 and any [lambda] < 2/3, a random formula with 'rn' 2-clauses and [lambda]' n' 3-clauses is a.s. satisfiable.