Mathematics

A finite non-increasing sequence of positive integers d=( d 1 ?�⋯??d n ) is called a degree sequence if there is a graph G=(V,E) with V={ v 1 ,?? v n } and deg( v i )= d i for i=1,??n . In that case we say that the graph G realizes the degree sequence d . We show that the exact number of labeled graphs that realize a fixed degree sequence satisfies a simple recurrence relation. Using this relation, we then obtain a recursive algorithm for the exact count. We also show that in the case of regular graphs the complexity of our algorithm is better than the complexity of the same enumeration that uses generating functions.

We enumerate the number of staircase diagrams over classically finite E -type Dynkin diagrams, extending the work of Richmond and Slofstra (Staircase Diagrams and Enumeration of smooth Schubert varieties) and completing the enumeration of staircase diagrams over finite type Dynkin diagrams. The staircase diagrams are in bijection to smooth and rationally smooth Schubert varieties over E -type thereby giving an enumeration of these varieties.

We show that one can enumerate the vertices of the convex hull of integer points in polytopes whose constraint matrices have bounded and nonzero subdeterminants, in time polynomial in the dimension and encoding size of the polytope. This extends a previous result by Artmann et al. who showed that integer linear optimization in such polytopes can be done in polynomial time.

Let G=(V,E) be a simple graph. A dominating set of G is a subset D?�V such that every vertex not in D is adjacent to at least one vertex in D . The cardinality of a smallest dominating set of G , denoted by γ(G) , is the domination number of G . A dominating set D is an accurate dominating set of G , if no |D| -element subset of V?�D is a dominating set of G . The accurate domination number, γ a (G) , is the cardinality of a smallest accurate dominating set D . In this paper, after presenting preliminaries, we count the number of accurate dominating sets of some specific graphs.

A Deza graph ? with parameters (v,k,b,a) is a k -regular graph with v vertices such that any two distinct vertices have b or a common neighbours, where b?�a . A Deza graph of diameter 2 which is not a strongly regular graph is called a strictly Deza graph. We find all 139 strictly Deza graphs up to 21 vertices.

The classical approach to envy-free division and equilibrium problems relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology. We illustrate the method by proving several relatives (extensions) of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.

The Erdos-Hajnal conjecture says that for every graph H there exists c>0 such that every graph G not containing H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. We prove that this is true when H is a cycle of length five. We also prove several further results: for instance, that if C is a cycle and H is the complement of a forest, there exists c>0 such that every graph G containing neither of C,H as an induced subgraph has a clique or stable set of cardinality at least |G|^c.

We study asymptotic properties of sequences of partitions ( ? \nobreakdash-algebras) in spaces with Bernoulli measures associated with the Robinson--Schensted--Knuth correspondence.

The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or not, colored or not. Let p?�[0,1] and q?�[0,1] be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-gamma-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly increasing, and so they are unimodal with modes in the middle, where fix and cyc are the fixed point and cycle statistics. When p=1 and q=1/2, we find a combinatorial interpretation of the bi-gamma-coefficients of the (p,q)-Eulerian polynomials. We then study excedance and flag excedance statistics of signed permutations and colored permutations. In particular, we establish the relationships between the (p,q)-Eulerian polynomials and some multivariate Eulerian polynomials. Our results unify and generalize a variety of recent results.

An {\em orthomorphism} over a finite field F is a permutation θ:F?�F such that the map x?��?x)?�x is also a permutation of F . The orthomorphism θ is {\em cyclotomic of index k } if θ(0)=0 and θ(x)/x is constant on the cosets of a subgroup of index k in the multiplicative group F ??. We say that θ has {\em least index} k if it is cyclotomic of index k and not of any smaller index. We answer an open problem due to Evans by establishing for which pairs (q,k) there exists an orthomorphism over F q that is cyclotomic of least index k . Two orthomorphisms over F q are orthogonal if their difference is a permutation of F q . For any list [ b 1 ,?? b n ] of indices we show that if q is large enough then F q has pairwise orthogonal orthomorphisms of least indices b 1 ,?? b n . This provides a partial answer to another open problem due to Evans. For some pairs of small indices we establish exactly which fields have orthogonal orthomorphisms of those indices. We also find the number of linear orthomorphisms that are orthogonal to certain cyclotomic orthomorphisms of higher index.