The small subgraph conditioning method first appeared when Robinson and the second author showed the almost sure hamiltonicity of random d-regular graphs. Since then it has been used to study the almost sure existence of, and the asymptotic distribution of, regular spanning subgraphs of various types in random d-regular graphs and hypergraphs. In this paper, we use the method to prove the almos...

tuczak, T., Holes in random graphs, Discrete Applied Mathematics 30 (1991) 213-219. It is shown that for every E>O with the probability tending to 1 as n-00 a random graph G(n,p) contains induced cycles of all lengths k, 3 <k~ (1 &)n log C/C, provided c(n) = (n l)p(n)-a.

In this short note, we prove a conjecture of Benjamini, Shinkar, and Tsur on the acquaintance time AC(G) of a random graph G ∈ G(n, p). It is shown that asymptotically almost surely AC(G) = O(log n/p) for G ∈ G(n, p), provided that pn−log n−log log n→∞ (that is, above the threshold for Hamiltonicity). Moreover, we show a matching lower bound for dense random graphs, which also implies that asym...

In this paper we introduce an improvement in the path planning algorithm for the humanoid soccer playing robot which uses Ferguson splines and PSO (Particle Swarm Optimization). The objective of the algorithm is to find a path through other playing robots to the ball, which should be as short as possible and also safe enough. Ferguson splines create preliminary paths using random generated para...