Ramsey Properties of Random Subgraphs of Pseudo-Random Graphs
نویسنده
چکیده
Let G = (V,E) be a d-regular graph of order n. Let Gp be the random subgraph of G for which each edge is selected from E(G) independently at random with probability p. For a fixed graph H, define m(H) := max{e(H ′)/(v(H ′)− 1) : H ′ ⊆ H}. We prove that n(m(H)−1)/m(H)/d is a threshold function for Gp to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex coloring, respectively, provided the eigenvalue λ of G that is second largest in absolute value is significantly smaller than d. As a consequence, it is also shown that n(m(H)−1)/m(H)/d is a threshold function for Gp to contain a family of vertex disjoint copies of H (an H-packing) that covers (1 − o(1))n vertices of G. Using a similar argument, the sharp threshold function for Gp to contain H as a subgraph is obtained as well.
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عنوان ژورنال:
- Electr. J. Comb.
دوره 19 شماره
صفحات -
تاریخ انتشار 2012