N. Alon and R. Yuster, Threshold functions for H-factors. Combin. Probab. Comput. 2 (1993), 137--144.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....= 1 o(1) provided p n 2 3 1 18 . Since G 3 (n 3, p) G(n, p) for 3 n, we have Corollary 1.7 If 3 n and p o(1) This improves the previous bound of Krivelevich [10] of p cn 2 3 1 15 . An earlier bound was c # n 2 3 1 6 (ln n) due to Rucinnski [13] and Alon and Yuster [1]. The obvious conjecture is still open. Conjecture. Let 3 n and the expected triangle degree # p = p Pr[G(n, p) has a perfect traingle packing] # if # p ##. If # p # #, it is easy to verify that with high probability there is a vertex contained in no triangle. Thus, ....

N. Alon and R. Yuster, Threshold functions for H-factors. Combin. Probab. Comput. 2 (1993), 137--144.

.... the existence of a vertex partition into triangles remains as one of the most challenging problems in this area (see, for example, the Appendix by ErdSs to the monograph by Alon and Spencer [1] The thresholds for H factors have been studied for example by Rucifiski [15] and by Alon and Yuster [3]. For a graph H, let ml(H) max( E(H) IV(H)l 1 where the maximum is taken over all subgraphs H of the graph H with at least two vertices. In [15] Rucifiski showed that the probability p(n) O(n 1Imp(H) is a sharp threshold for the property C,l for any graph H such that ml(H) H) where ....

N. Alon, R. Yuster, Threshold functions for H-factors, Combinatorics, Probability, & Computing 2 (1993), 137-144.

....H factor For the case H = K 2 the solution has been given by Erd os and R enyi in [3] they showed that p = O(log n=n) suces to have whp a perfect matching in G 2 G(n; p) For the case of a general graph H the problem remains unsolved. Some partial results have been obtained by Alon and Yuster [2] and by Ruci nski [5] However, the problem is still open for many important classes of graphs, in particular, for the case H = K r for every r 3. In this paper we are mostly concerned with the case H = K 3 , this graph is actually the smallest one for which the problem is yet unsolved. ....

N. Alon and R. Yuster, Threshold functions for H-factors, Comb., Prob. and Computing 2 (1993), 137-144.

.... the existence of a vertex partition into triangles remains as one of the most challenging problems in this area (see, for example, the Appendix by Erd os to the monograph by Alon and Spencer [1] The thresholds for H factors have been studied for example by Ruci nski [15] and by Alon and Yuster [3]. For a graph H; let m 1 (H) max jE(H 0 )j jV (H 0 )j 1 where the maximum is taken over all subgraphs H 0 of the graph H with at least two vertices. In [15] Ruci nski showed that the probability p(n) O(n 1=m1 (H) is a sharp threshold for the property CH;1 for any graph H ....

N. Alon, R. Yuster, Threshold functions for H-factors, Combinatorics, Probability, & Computing 2 (1993), 137-144.

....For the case H = K 2 the solution has been given by Erdos and R enyi in [3] they showed that p = O(log n=n) suffices to have whp a perfect matching in G 2 G(n; p) For the case of a general graph H the problem remains unsolved. Some partial results have been obtained by Alon and Yuster [2] and by Ruci nski [5] However, the problem is still open for many important classes of graphs, in particular, for the case H = K r for every r 3. In this paper we are mostly concerned with the case H = K 3 , this graph is actually the smallest one for which the problem is yet unsolved. This ....

N. Alon and R. Yuster, Threshold functions for H-factors, Comb., Prob. and Computing 2 (1993), 137--144.

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