$\delta$-Connectivity in Random Lifts of Graphs

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.

Download Full-text

Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

ISRN Combinatorics ◽

10.1155/2013/806583 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17

Author(s):

Sergey Kitaev ◽

Jeffrey Remmel ◽

Manda Riehl

Keyword(s):

Symmetric Group ◽

Wreath Product ◽

Cyclic Group ◽

Generating Functions ◽

Symmetric Groups ◽

Pattern Avoidance ◽

Catalan Numbers ◽

Cyclic Groups ◽

Bijective Proof ◽

Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

Download Full-text

Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000259 ◽

2020 ◽

pp. 1-7

Author(s):

Omar Tout

Keyword(s):

Finite Group ◽

Symmetric Group ◽

Wreath Product ◽

Abelian Groups ◽

Symmetric Groups ◽

Gelfand Pair ◽

Gelfand Pairs ◽

Finite Abelian Groups ◽

A Finite Group

Abstract It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

Download Full-text

The center of the wreath product of symmetric group algebras

Algebra and Discrete Mathematics ◽

10.12958/adm1338 ◽

2021 ◽

Vol 31 (2) ◽

pp. 302-322

Author(s):

O. Tout ◽

Keyword(s):

Symmetric Group ◽

A Sharp Improvement of a Theorem of Bauer and Schmeichel

Mathematical Problems of Computer Science ◽

10.51408/1963-0017 ◽

2018 ◽

pp. 15-34

Author(s):

Zhora Nikoghosyan

Keyword(s):

Lower Bound ◽

Connected Graph ◽

Classical Result ◽

Minimum Degree

Let G be a graph on n vertices with minimum degree δ. The earliest nontrivial lower bound for the circumference c (the length of a longest cycle in G) was established in 1952 due to Dirac in terms of n and δ: (i) if G is a 2-connected graph, then c ≥ min{n, 2δ}. The bound in Theorem (i) is sharp. In 1986, Bauer and Schmeichel gave a version of this classical result for 1-tough graphs: (ii) if G is a 1-tough graph, then c ≥ min{n, 2δ + 2}. In this paper we present an improvement of (ii), which is sharp for each n: (iii) if G is a 1-tough graph, then c ≥ min{n, 2δ + 2} when n ≡ 1(mod 3); c ≥ min{n, 2δ + 3} when n ≡ 2(mod 3) or n ≡ 1(mod 4); and c ≥ min{n, 2δ + 4} otherwise.

Download Full-text

On the transfer in the homology of symmetric groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100054323 ◽

1978 ◽

Vol 83 (1) ◽

pp. 91-101 ◽

Cited By ~ 10

Author(s):

Daniel S. Kahn ◽

Stewart B. Priddy

Keyword(s):

Symmetric Group ◽

Wreath Product ◽

Homotopy Theory ◽

Symmetric Groups ◽

Characterization Theorem ◽

Group Cohomology ◽

Stable Homotopy ◽

Stable Homotopy Theory ◽

Transfer Homomorphism ◽

Image Position

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphismin group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

Download Full-text

A Note on Conditional Edge Connectivity of Hypercube Networks

Journal of Interconnection Networks ◽

10.1142/s0219265921500079 ◽

2021 ◽

pp. 2150007

Author(s):

J. B. Saraf ◽

Y. M. Borse

Keyword(s):

Lower Bound ◽

Connected Graph ◽

Minimum Degree ◽

Edge Connectivity ◽

Hypercube Networks ◽

Sharp Lower Bound

Let [Formula: see text] be a connected graph with minimum degree at least [Formula: see text] and let [Formula: see text] be an integer such that [Formula: see text] The conditional [Formula: see text]-edge ([Formula: see text]-vertex) cut of [Formula: see text] is defined as a set [Formula: see text] of edges (vertices) of [Formula: see text] whose removal disconnects [Formula: see text] leaving behind components of minimum degree at least [Formula: see text] The characterization of a minimum [Formula: see text]-vertex cut of the [Formula: see text]-dimensional hypercube [Formula: see text] is known. In this paper, we characterize a minimum [Formula: see text]-edge cut of [Formula: see text] Also, we obtain a sharp lower bound on the number of vertices of an [Formula: see text]-edge cut of [Formula: see text] and obtain some consequences.

Download Full-text

A New Bound on the Domination Number of Graphs with Minimum Degree Two

The Electronic Journal of Combinatorics ◽

10.37236/499 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

Michael A. Henning ◽

Ingo Schiermeyer ◽

Anders Yeo

Keyword(s):

Connected Graph ◽

Minimum Degree ◽

Domination Number ◽

Degree Condition ◽

Cut Vertex ◽

Delta G ◽

Vertex Disjoint ◽

Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

Download Full-text

On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

The Electronic Journal of Combinatorics ◽

10.37236/5173 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 8

Author(s):

Jakub Przybyło

Keyword(s):

Connected Graph ◽

Special Class ◽

Maximum Degree ◽

Minimum Degree ◽

Probabilistic Approach ◽

The Family ◽

Edge Colourings ◽

Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

Download Full-text

Certain unitary representations of the infinite symmetric group, II

Nagoya Mathematical Journal ◽

10.1017/s0027763000000933 ◽

1987 ◽

Vol 106 ◽

pp. 143-162 ◽

Cited By ~ 4

Author(s):

Nobuaki Obata

Keyword(s):

Symmetric Group ◽

Discrete Group ◽

Inductive Limit ◽

Symmetric Groups ◽

Discrete Groups ◽

Type I ◽

Infinite Symmetric Group ◽

Locally Finite ◽

Distinctive Features ◽

Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

Download Full-text

The multistep homology of the simplex and representations of symmetric groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004119000124 ◽

2019 ◽

Vol 169 (2) ◽

pp. 231-253

Author(s):

MARK WILDON

Keyword(s):

Symmetric Group ◽

Symmetric Groups ◽

Explicit Construction ◽

Fixed Size ◽

Simplicial Homology ◽

Chain Complexes ◽

Basic Spin ◽

Characteristic Two

AbstractThe symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

Download Full-text