Nicholas C. Wormald

Sudden Emergence of a Giantk-Core in a Random Graph

Thek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdo?s?R?nyi random graphG(n,?m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton/2. We show that fork?3, with high probability, a giantk-core appears suddenly whenmreachesckn/2; hereck=min?>0?/?k(?) and?k(?)=P{Poisson(?)?k?1}. In particular,c3?3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ?pk(?k)n. Here?kis the minimum point of?/?k(?) andpk(?k)=P{Poisson(?k)?k}. Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one.

Edge crossings in drawings of bipartite graphs

Systems engineers have recently shown interest in algorithms for drawing directed graphs so that they are easy to understand and remember. Each of the commonly used methods has a step which aims to adjust the drawing to decrease the number of arc crossings. We show that the most popular strategy involves an NP-complete problem regarding the minimization of the number of arcs in crossings in a bipartite graph. The performance of the commonly employed “barycenter” heuristic for this problem is analyzed. An alternative method, the “median” heuristic, is proposed and analyzed. The new method is shown to compare favorably with the old in terms of performance guarantees. As a bonus, we show that the median heuristic performs well with regard to the total length of the arcs in the drawing.

Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2)

AbstractWe determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|1/3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|1/4). In particular, ifk=o(n1/2), the number of regular graphs of degreek and ordern is asymptotically

Uniform generation of random regular graphs of moderate degree

\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).

Almost all cubic graphs are Hamiltonian

Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.

The asymptotic distribution of short cycles in random regular graphs

In a previous article the authors showed that almost all labelled cubic graphs are hamiltonian. In the present article, this result is used to show that almost all r‐regular graphs are hamiltonian for any fixed r ⩾ 3, by an analysis of the distribution of 1‐factors in random regular graphs. Moreover, almost all such graphs are r‐edge‐colorable if they have an even number of vertices. Similarly, almost all r‐regular bipartite graphs are hamiltonian and r‐edge‐colorable for fixed r ⩾ 3.

Birth control for giants

Abstract We show how to generate k -regular graphs on n vertices uniformly at random in expected time O ( nk 3 ), provided k = O(n 1 3 ) . The algorithm employs a modification of a switching argument previously used to count such graphs asymptotically for k = o(n 1 3 ) . The asymptotic formula is re-derived, using the new switching argument. The method is applied also to graphs with given degree sequences, provided certain conditions are met. In particular, it applies if the maximum degree is O(∥E(G)∥ 1 4 ) . The method is also applied to bipartite graphs.

Asymptotic enumeration by degree sequence of graphs of high degree

We present a practical algorithm for generating random regular graphs. For all d growing as a small power of n, the d-regular graphs on n vertices are generated approximately uniformly at random, in the sense that all d-regular graphs on n vertices have in the limit the same probability as n → ∞. The expected runtime for these ds is O(nd2).