Diestel Graph Theory

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40) give lim Pr(RG,,+,,, n-+ m E ~cgllVI(RG,,+,,,)l=n)=l-e-e-4cls. B. Bollobds, A . M . 6. 23) we obtain where s,(nz’) denotes the expected number of vertices of degree 1 in a random graph chosen uniformly from %,(m’). 28). 47). 0 The reader familiar with [I] will realize that we had to work with multigraphs and proceed in this way because the probability that a graph in Q(d) has no loops or multiple edges is too small. 1. 30). For c,,+ + co, c,ldlogn we simply repeat the arguments almost unchanged.

W. Robinson and N. C . Wormald, On hamiltonian cycles in 3-connected cubic maps, Annals of Discrctc Mathematics (to appear). [I51 L. B. Richmond and N . C. Worniald, On random triangulations of thc plane. [16] R. W. Robinson and N. C. Wormald, Almost all bipartite cubic graphs are hamiltonian. [17] E. Shamir, How many random edge3 make a graph hamiltonian, Combinatorica 3 (1983) 123-132. [18] E. Shamir and E. Upfal, On factors in random gruphs, Israel Journal of Mathematics 39(1981)296-302. [19] E.

2. I lc) Now consider the components of. the graph induced by the isolated veitices I ofH[S,] and their neighbors in G. Let C be the set ofvertices ofsuch a component. lc), that I C n T I G 1 . ( 7 . 1 Id) To deal with the case IC n 1132 we note t h a t if u, u E I then by (2. 1~ ) N , ( { u ) ) n NG({u})#O implies N , , ( { u ) )n ' T ' = @ . 10). 9). Case 1. >21s21. le) we obtain \s,I Case 2. IS, I < 2 1 ~ , l <2n/Iog n. From (2. 10) we have I Cuse 3. IS1 <21 Szl, r1/logt1< lszl6n/8000. le).

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