Matching and covering the vertices of a random graph by copies of a given graph
Discrete Mathematics
On fractional K-factors of random graphs
Random Structures & Algorithms
Thresholds and Expectation Thresholds
Combinatorics, Probability and Computing
Random Structures & Algorithms
Increasing the flexibility of the herding attack
Information Processing Letters
Research paper: Combinatorial and computational aspects of graph packing and graph decomposition
Computer Science Review
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For a graph G=(V, E) on n vertices, where 3 divides n, a triangle factor is a subgraph of G, consisting of n/3 vertex disjoint triangles (complete graphs on three vertices). We discuss the problem of determining the minimal probability p=p(n), for which a random graph G∈𝒢(n, p) contains almost surely a triangle factor. This problem (in a more general setting) has been studied by Alon and Yuster and by Ruciński, their approach implies p=O((log n/n)1/2). Our main result is that p=O(n)−3/5) already suffices. The proof is based on a multiple use of the Janson inequality. Our approach can be extended to improve known results about the threshold for the existence of an H-factor in 𝒢(n, p) for various graphs H.