Subgraphs of Random Graphs with Specified Degrees

McKay, Brendan D.

doi:10.1142/9789814324359_0155

Cited by 64 publications

(80 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As for random d-regular graphs with d = ω(1), the Hamiltonicity [7], the connectivity [21], the number of specified subgraphs [16,17,12] and many other properties have been explored [21]. However, the diameter has not been known explicitly.…”

Section: Theorem 12 Fix Two Vertices S and T Letmentioning

confidence: 99%

“…For l = l(n) ∈ N, we consider the number of paths of length l connecting two fixed vertices s and t contained in G n,d . The following theorem due to McKay [16] is useful.…”

Section: Subgraph Counting Techniquementioning

confidence: 99%

“…The algorithm is based on switching method that can be used to analyze G n,d with d = ω (1). Since their work, the switching method has been used to explore several properties of G n,d including the Hamiltonicity [7], the connectivity [9], the number of subgraphs of specific type [12,16,17] (see [21] for details). However, to the best of our knowledge, the diameter has not been known explicitly.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Diameter of Dense Random Regular Graphs

Shimizu

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

There is a tight upper bound on the order (the number of vertices) of any d-regular graph of diameter D, known as the Moore bound in graph theory. This bound implies a lower bound D 0 (n, d) on the diameter of any d-regular graph of order n. Actually, the diameter diam (G n,d ) of a random dregular graph G n,d of order n is known to be asymptotically "optimal" as n → ∞. Bollobás and de la Vega (1982) holds w.h.p. (with high probability) for fixed d ≥ 3, whereas there exists a gapIn this paper, we investigate the gap diam( (1)) n α where α ∈ (0, 1) and β > 0 are arbitrary constants. We prove that diam(G n,d ) = ⌊α −1 ⌋ + 1 holds w.h.p. for such d. Our result implies that the gap is 1 if α −1 is an integer and d ≥ n α , and is 0 otherwise. One can easily obtain that diam (G n,d ) ≤ ⌊α −1 ⌋ + 1 holds w.h.p. by using the embedding theorem due to Dudek et al. (2017). Our critical contribution is to show that diam (G n,d ) ≥ ⌊α −1 ⌋ + 1 holds w.h.p. by the analysis of the distances of fixed vertex pairs.

show abstract

Section: Theorem 12 Fix Two Vertices S and T Letmentioning

confidence: 99%

“…For l = l(n) ∈ N, we consider the number of paths of length l connecting two fixed vertices s and t contained in G n,d . The following theorem due to McKay [16] is useful.…”

Section: Subgraph Counting Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Diameter of Dense Random Regular Graphs

Shimizu

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…Another technique proven to be useful when dealing with random regular graphs is the so called edge switching technique introduced to the random regular graph scene by McKay in [32] in order to obtain general bounds on the probability of a subgraph occurrence. Another, more modern kind of switchings, was introduced by McKay and Wormald in [34] and [35], and this is the kind we apply here.…”

Section: Random Regular Graphsmentioning

confidence: 99%

The logic of random regular graphs

Haber

Krivelevich

2010

Journal of Combinatorics

View full text Add to dashboard Cite

The first order language of graphs is a formal language in which one can express many properties of graphs -known as first order properties. The classic Zero-One law for random graphs states that if p is some constant probability then for every first order property the limiting probability of the binomial random graph G(n, p) having this property is either zero or one. The case of sparse random graphs has also been studied in detail for the binomial random graph model. We obtain results for random regular graphs that match the main results for G (n, p). In particular we prove that if the degree d is linear in the number of vertices n,obeys the Zero-One law. On the contrary, if d = n α for rational 0 < α < 1, then there is a (theoretically explicit) first order property with no limiting probability in G n,d .

show abstract

“…Moreover, whenever there are objects Q ∈ C(v), R ∈ C(w) such that Q can be taken onto R by a switching, there is a directed edge (v, w) in E. There are many examples in the literature where classes of combinatorial objects are approximately enumerated by this technique in the case that G is a directed path. A few examples are [1,3,6,7,8,9,10,11]. Fack and McKay [2] gave a more general analysis, allowing G to be an arbitrary acyclic directed graph, plus optional loops.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial estimates by the switching method

Hasheminezhad¹,

McKay²

2010

Combinatorics and Graphs

Self Cite

View full text Add to dashboard Cite

Abstract. The method of switchings is a standard tool for enumerative and probabilistic applications in combinatorics. In its simplest form, it analyses a relation between two sets to estimate the ratio of their sizes. In a more complicated setting, there is a family of sets connected by some relations. By bounding properties of the relations, bounds can be inferred on the relative sizes of the sets. In this paper we extend the treatment of Fack and McKay (2007) to allow the graph of sets and relations to be an arbitrary directed graph. A special case that frequently occurs in bounding tails of distributions is analysed in detail.

show abstract

Subgraphs of Random Graphs with Specified Degrees

Cited by 64 publications

References 29 publications

The Diameter of Dense Random Regular Graphs

The Diameter of Dense Random Regular Graphs

The logic of random regular graphs

Combinatorial estimates by the switching method

Contact Info

Product

Resources

About