André E. Kézdy

Partitioning permutations into increasing and decreasing subsequences

A permutation is an (r, s)-permutation if it can be partitioned into r increasing and s decreasing, possibly empty subsequences. For any fixed non-negative integers r and s, the family of (r, s)-permutations is characterized by a finite list of forbidden subsequences. This is derived from a more general graph-theoretic proof showing that, for any fixed non-negative integers r and s, the family of perfect graphs whose vertex set admits a partition into r cliques and s independent sets if characterized by a finite list of forbidden induced subgraphs.

Distinct Sums Modulo n and Tree Embeddings

In this paper we are concerned with the following conjecture.Conjecture. For any positive integers n and k satisfying k < n, and any sequence a1, a2, … ak of not necessarily distinct elements of Zn, there exists a permutation π ∈ Sk such that the elements aπ(i)+i are all distinct modulo n.We prove this conjecture when 2k ≤ n+1. We then apply this result to tree embeddings. Specifically, we show that, if T is a tree with n edges and radius r, then T decomposes Kt for some t ≤ 32(2r+4)n2+1.

Clique covering the edges of a locally cobipartite graph

Abstract We prove that a locally cobipartite graph on n vertices contains a family of at most n cliques that cover its edges. This is related to Opsuts conjecture that states the competition number of a locally cobipartite graph is at most two.

The poset on connected induced subgraphs of a graph need not be Sperner

SupposeG is a finite connected graph. LetC(G) denote the inclusion ordering on the connected vertex-induced subgraphs ofG. Penrice asked whetherC(G) is Sperner for general graphsG. Answering Penrices question in the negative, we present a treeT such thatC(T) is not Sperner. We also construct a related distributive lattice that is not Sperner.

The 2-intersection number of paths and bounded-degree trees

We represent a graph by assigning each vertex a finite set such that vertices are adjacent if and only if the corresponding sets have at least two common elements. The 2-intersection number θ2(G) of a graph G is the minimum size of the union of sets in such a representation. We prove that the maximum order of a path that can be represented in this way using t elements is between (t2 - 19t + 4)/4 and (t2 - t + 6)/4, making θ2(Pn) asymptotic to 2√n. We also show the existence of a constant c depending on ϵ such that, for any tree T with maximum degree at most d, θ2(T) ≤ c(√n)1+ϵ. When the maximum degree is not bounded, there is an n-vertex tree T with θ2(T) > .945n2/3.

Polynomials that Vanish on Distinct

Let

n

{\bf C}

th Roots of Unity

denote the field of complex numbers and

A finite basis characterization of α-split colorings

\Omega_n

Alternating walks in partially 2-edge-colored graphs and optimal strength of graph labeling

the set of