Large Vertex-Disjoint Cycles in a Bipartite Graph

Let s≥2 and k be two positive integers. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n≥s k and the minimum degree at least (s−1)k+1. When s=2 and n >2k, it is proved in [5] that G contains k vertex-disjoint cycles. In this paper, we show that if s≥3, then G contains k vertex-disjoint cycles of length at least 2s. Received: March 2, 1998 Revised: October 26, 1998     

Partition conditions and vertex-connectivity of graphs

It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev ₁, ...,v _n ≠V(G) and for any sequence of positive integersk ₁, ...,k _n such thatk ₁+...+k _n =|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv ₁, ...,v(n), and the class of the partition containingv _i induces a connected subgraph consisting ofk _i vertices, fori=1, 2, ...,n. Now fix the integersk ₁, ...,k _n . In this paper we study what can we say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv ₁, ...,v _n ≠V(G). We find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases we give sharp lower bounds for the vertex-connectivity ofG.     

On 2‐factors of a bipartite graph

In this article, we consider the following problem: Given a bipartite graph G and a positive integer k, when does G have a 2‐factor with exactly k components? We will prove that if G = (V₁, V₂, E) is a bipartite graph with |V₁| = |V₂| = n ≥ 2k + 1 and δ (G) ≥ ⌈n/2⌉ + 1, then G contains a 2‐factor with exactly k components. We conjecture that if G = (V₁, V₂; E) is a bipartite graph such that |V₁| = |V₂| = n ≥ 2 and δ (G) ≥ ⌈n/2⌉ + 1, then, for any bipartite graph H = (U₁, U₂; F) with |U₁| ≤ n, |U₂| ≤ n and Δ (H) ≤ 2, G contains a subgraph isomorphic to H. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 101–106, 1999     

Maximal Total Length Of k Disjoint Cycles In Bipartite Graphs

Let k, s and n be three integers with sk2, n2k+1. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n. If the minimum degree of G is at least s+1, then G contains k vertex-disjoint cycles covering at least min(2n,4s) vertices of G.     

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V₁ and V₂ such that in G[V₁] every vertex has degree at most k, while G[V₂] is edgeless. For every integer k?0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x ₁, x ₂, . . . , x _n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x ₁, x ₂, . . . , x _n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.     

Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ⁺(G) = max{d⁺(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)² + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998     

Lie-admissible algebras and generalized witt algebras ∗

The symmetric group 𝔖_n+1 of degree n + 1 admits an n-dimensional irreducible Q𝔖_n-module V corresponding to the hook partition (2, 1^n?1). By the work of Craig and Plesken, we know that there are σ(n + 1) many isomorphism classes of Z𝔖_n+1-lattices which are rationally equivalent to V, where σ denotes the divisor counting function. In the present article, we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the Z𝔖_n+1-lattice defined by the Specht basis.     

On Independent Cycles in a Bipartite Graph

 Let G=(V ₁,V ₂;E) be a bipartite graph with 2k≤m=|V ₁|≤|V ₂|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs. Received: December 9, 1998 Final version received: June 2, 1999     

Permutations with Low Discrepancy Consecutive k-sums

Consider the permutation π=(π₁,…, π_n) of 1,2,…, n as being placed on a circle with indices taken modulo n. For given kn there are n sums of k consecutive entries. We say the maximum difference of any consecutive k-sum from the average k-sum is the discrepancy of the permutation. We seek a permutation of minimum discrepancy. We find that in general the discrepancy is small, never more than k+6, independent of n. For g= gcd(n,k)>1, we show that the discrepancy is . For g=1 it is more complicated. Our constructions show that the discrepancy never exceeds k/2 by more than 9 for large n, while it is at least k/2 for infinitely many n.We also give an analysis for the easier case of linear permutations, where we view the permutation as written on a line. The analogous discrepancy is at most 2 for all n,k.     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.     

A Technique for the Search for Vertices of the Components of a k G-Permutation Module

In this paper we present a necessary condition for a p-group V ≤ G to be a vertex of some indecomposable direct summand of the permutation module k _H  ↑ ^G , where H ≤ G, and G is a finite group. We call this condition H-suitability and present a method how to check for it. In an example, we determine all H-suitable groups. In fact, in this example every H-suitable group is the vertex of some indecomposable direct summand of k _H  ↑ ^G .     

The average Steiner distance of a graph

The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n-distance μ_n(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μ_n(G) ≤ μ_k(G) + μ_n+1−k(G) where 2 ≤ k ≤ n − 1. The range for the average Steiner n-distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n − 1, it is shown that μ_n(T) ≤ (n/k)μ_k(T) and the range for μ_n(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n-distance of a tree are outlined. © 1996 John Wiley & Sons, Inc.     

Cycles containing all vertices of maximum degree

For a graph G and an integer k, denote by V_k the set {v ∈ V(G) | d(v) ≥ k}. Veldman proved that if G is a 2-connected graph of order n with n ≤ 3k - 2 and |V_k| ≤ k, then G has a cycle containing all vertices of V_k. It is shown that the upper bound k on |V_k| is close to best possible in general. For the special case k = δ(G), it is conjectured that the condition |V_k| ≤ k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n ≤ 2δ(G) + δ(G) + 1. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n ≤ 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented. © 1993 John Wiley & Sons, Inc.     

On the Grassmann modules for the unitary groups

Let V be 2n-dimensional vector space over a field 𝕂 equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let 𝕂₀ ? 𝕂 be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, …, 2n}, there exist natural representations of the groups G ? U(2n, 𝕂/𝕂₀) and H = G ∩ SL(V) ? SU(2n, 𝕂/𝕂₀) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

Enumeration of totally positive Grassmann cells

Postnikov (Webs in totally positive Grassmann cells, in preparation) has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr_k,n⁺, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in Gr_k,n⁺ according to their dimension. As a corollary, we give a new proof that the Euler characteristic of Gr_k,n⁺ is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.     

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph K_m(n₁,n₂,…,n_m) with vertex set V and m independent sets G₁,G₂,…,G_m of n₁,n₂,…,n_m vertices respectively. Let G={G₁,G₂,…,G_m}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?G_i for some G_i∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type g^u. In this paper, we show that there exists a (k,λ)-cycle frame of type g^u for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.