Goldberg's conjecture is true for random multigraphs

Haxell, Penny; Krivelevich, Michael; Kronenberg, Gal

doi:10.1016/j.jctb.2019.02.005

Cited by 3 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is well‐known and easy to see that G m ∼ G ( n , m ) for every

0 \leq m \leq ((), \binom{n}{2})

, that is, this random graph process generates the Erdős‐Renyi random graph model . The second process we consider is the random multigraph process { M ( n , m )} m ≥ 0 that was introduced in . This process is similar to the first one except that, in each round, the edge we add is chosen u.a.r.…”

Section: Introductionmentioning

confidence: 99%

“…The (possibly randomized) algorithm that Builder uses in order to add edges throughout this process is called the strategy of Builder. As a special case, we also show how the process can be used to approximate (using suitable strategies) some well‐known random graph models such as the Erdős‐Renyi random graph model , the random multigraph model , the k ‐out model , and the min‐degree process .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Semi‐random graph process

Ben‐Eliezer

Hefetz

Kronenberg

et al. 2019

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties 𝒫, we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies 𝒫. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well‐studied random graph models.

show abstract

“…It is well‐known and easy to see that G m ∼ G ( n , m ) for every

0 \leq m \leq ((), \binom{n}{2})

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semi‐random graph process

Ben‐Eliezer

Hefetz

Kronenberg

et al. 2019

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, it is also worth mentioning that very recently, Haxell, Krivelevich and Kronenberg [17] studied a related problem in a random multigraph setting; it is interesting to check whether our techniques can be applied there as well.…”

Section: Random Graphsmentioning

confidence: 99%

1‐Factorizations of pseudorandom graphs

Ferber

Jain

2020

Random Struct Algorithms

View full text Add to dashboard Cite

A 1-factorization of a graph G is a collection of edge-disjoint perfect matchings whose union is E(G). In this paper, we prove that for any > 0, an (n, d, λ)-graph G admits a 1-factorization provided that n is even, C 0 ≤ d ≤ n − 1 (where C 0 = C 0 ( ) is a constant depending only on ), and λ ≤ d 1− . In particular, since (as is well known) a typical random d-regular graph G n,d is such a graph, we obtain the existence of a 1-factorization in a typical G n,d for all C 0 ≤ d ≤ n−1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G, which is better by a factor of 2 nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.

show abstract

“…It is also worth mentioning that very recently, Haxell, Krivelevich and Kronenberg [15] studied a related problem in a random multigraph setting; it is interesting to check whether our techniques can be applied there as well.…”

Section: Almost All D-regular Graphs Are Of Classmentioning

confidence: 99%

1-Factorizations of Pseudorandom Graphs

Ferber

Jain

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

A 1-factorization of a graph G is a collection of edge-disjoint perfect matchings whose union is E(G). A trivial necessary condition for G to admit a 1-factorization is that |V (G)| is even and G is regular; the converse is easily seen to be false. In this paper, we consider the problem of finding 1-factorizations of regular, pseudorandom graphs. Specifically, we prove that an (n, d, λ)graph G (that is, a d-regular graph on n vertices whose second largest eigenvalue in absolute value is at most λ) admits a 1-factorization provided that n is even, C 0 ≤ d ≤ n − 1 (where C 0 is a universal constant), and λ ≤ d 0.9 . In particular, since (as is well known) a typical random dregular graph G n,d is such a graph, we obtain the existence of a 1-factorization in a typical G n,d for all C 0 ≤ d ≤ n − 1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G which is off by a factor of 2 in the base of the exponent from the known upper bound. This lower bound is better by a factor of 2 nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph. Our proofs are probabilistic and can be easily turned into polynomial time (randomized) algorithms.

show abstract

Goldberg's conjecture is true for random multigraphs

Cited by 3 publications

References 29 publications

Semi‐random graph process

Semi‐random graph process

1‐Factorizations of pseudorandom graphs

1-Factorizations of Pseudorandom Graphs

Contact Info

Product

Resources

About