Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $\lambda_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le \epsilon_n0$, a random induced subgraph has a.s. a unique largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where $\wp(\epsilon_n)$ is the survival probability of a specific branching process.Comment: 18 pages, 1 figur

Routing Algorithms on the Bus-Based Hypercube Network

Abstract—In this paper, we study the properties of the bus-based hypercube, denoted as Uðn; bÞ, which is a kind of multiple-bus networks (MBN). Uðn; bÞ consists of 2n processors and 2b buses, where 0 b n 1, and each processor is connected to either dbþ2 2 e or dbþ1 2 e buses. We show that the diameter of Uðn; bÞ is dbþ1 2 e if b 2. We also present an algorithm to select the best neighbor processor via which we can obtain one shortest routing path. In Uðn; bÞ, we show that if there exist some faults, the fault diameter b 3 DFðn; b; fÞ b þ 1, where f is the sum of bus faults and processor faults and 0 f d 2 e. Furthermore, we also show that the busfault diameter DBðn; b; fÞ bb 1 2cþ3, where 0 f db 2 e and f is the number of bus faults. These results improve significantly the previous result that DBðn; b; fÞ b þ 2f þ 1, where f is the number of bus faults. Index Terms—Multiple-bus network, hypercube, routing algorithm, diameter, fault tolerance.