The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version apart from formatting and a minor amendment to Lemma 8 (and its proof on p. 21

The k-core and branching processes

The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold $\lambda_c$ for the emergence of a non-trivial k-core in the random graph $G(n,\lambda/n)$, and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k-core in a certain power-law or `scale-free' graph with a parameter c controlling the overall density of edges. For each k at least 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k-core when c is \epsilon above the threshold. In contrast to $G(n,\lambda/n)$, this fraction tends to 0 as \epsilon tends to 0.Comment: 30 pages, 1 figure. Minor revisions. To appear in Combinatorics, Probability and Computin

Determination of a Graph\u27s Chromatic Number for Part Consolidation in Axiomatic Design

Mechanical engineering design practices are increasingly moving towards a framework called axiomatic design (AD). A key tenet of AD is to decrease the information content of a design in order to increase the chance of manufacturing success. An important way to decrease information content is to fulfill multiple functional requirements (FRs) by a single part: a process known as part consolidation. One possible method for determining the minimum number of required parts is to represent a design by a graph, where the vertices are the FRs and the edges represent the need to separate their endpoint FRs into separate parts. The answer is then the chromatic number of such a graph. This research investigates the suitability of using two existing algorithms and a new algorithm for finding the chromatic number of a graph in a part consolidation tool that can be used by designers. The runtime complexities and durations of the algorithms are compared empirically using the results from a random graph analysis with binomial edge probability. It was found that even though the algorithms are quite different, they all execute in the same amount of time and are suitable for use in the desired design tool