Yair Caro

Improved lower bounds on k-independence

A vertex set Y in a (hyper)graph is called k-independent if in the sub(hyper)-graph induced by Y every vertex is incident to less than k edges. We prove a lower bound for the maximum cardinality of a k-independent set—in terms of degree sequences—which strengthens and generalizes several previously known results, including Turans theorem.

Combinatorial reconstruction problems

Abstract A general technique for tackling various reconstruction problems is presented and applied to some old and some new instances of such problems.

On the largest tree of given maximum degree in a connected graph

We prove that every connected graph G contains a tree T of maximum degree at most k that either spans G or has order at least kδ(G) + 1, where δ(G) is the minimum degree of G. This generalizes and unifies earlier results of Bermond [1] and Win [7]. We also show that the square of a connected graph contains a spanning tree of maximum degree at most three.

Generalized 1-factorization of trees

Abstract Trees having a 1-factor and trees having a generalized 1-factorization are characterized.

Decompositions of partially ordered sets into chains and antichains of given size

Every partially ordered set P on at least (1+o(1))n3 elements can be decomposed into subposets of size n that are ‘almost’ chains or antichains. This lower bound on P is asymptotically best possible. Similar results are presented for other types of combinatorial structures.

Bisection of trees and sequences

Alon, N., Y. Caro and I. Krasikov, Bisection of trees and sequences, Discrete Mathematics 114 (1993) 337. A graph G is called bisectable if it is an edge-disjoint union of two isomorphic subgraphs. We show that any tree T with e edges contains a bisectable subgraph with at least e - O(e/log log e) edges. We also show that every forest of size e, each component of which is a star, contains a bisectable subgraph of size at least e -O(log2 e). Let G be a graph with n = n(G) vertices and e = e(G) edges. The number of edges e of G is called the size of G. G is bisectab2e if it is an edge-disjoint union of two isomorphic subgraphs. Let B(G) be a bisectable subgraph of maximum size of G. The function R(G) = e(G) -e(B(G)) for general graphs G has been studied by ErdBs et al. [l] and independently by Alon and Krasikov (unpublished). It was shown that any graph of size e contains a bisectable subgraph with at least R(e2j3) edges, and that there are graphs of size e containing no bisectable subgraphs of size more than 0(e2/310g e/log log e). Here we consider the function R(G) in two special cases; when G is a tree and when G is a forest, each connected component of which is a star. Some other results dealing with decompositions of trees into isomorphic subgraphs appear in [4] and in some of its references.

Extremal problems concerning transformations of the set of edges of the complete graph

Let E m denote the set of edges of the complete graph on m vertices K m , and let f : E m → E m be a function. A subgraph G = ( V ( G ), E ( G )) of K m is called f -fixed if f ( e ) = e for all e ∈ E ( G ) and f -free if f ( e ) ∉ E ( G ) for all e ∈ E ( G ). For two finite graphs G , H we define m ( G , H ) = max { m : ∃ f : E m → E m such that no copy of G in K m is f - fixed and no copy of H in K m is f - free } If m > 2 we define l ( m , H ) = max { l : ∃ f : E m → E m , f ( e ) ≠ e for l edges e ∈ E m and no copy of H in K m is f - free } In this paper we investigate the functions m ( G , H ) and l ( m , H ). We determine m ( G , H ) precisely for some families of graphs and estimate the asymptotic behaviour of l ( m , H ) for fixed H as m tends to infinity. Some of the results are generalised to functions defined on the set of edges of a hypergraph.

On a division property of consecutive integers

Pillai and Brauer proved that form≧17 we can find blocksBm ofm consecutive integers such that no element in the block is pairwise prime with each of the other elements. The following basic generalization is proved: For eachd>1 there is a numberG(d) such that for everym≧G(d) there exist infinitely many blocksBm ofm consecutive integers, such that for eachr∈Bm there existss∈Bm, (r,s)≧d.

Decomposition of large combinatorial structures

Etude de la decomposition de grandes structures combinatoires. Application aux hypergraphes

Extremal problems concerning transformations of the edges of the complete hypergraphs

We consider extremal problems concerning transformations of the edges of complete hypergraphs. We estimate the order of the largest subhypergraph K such that for every edge e ϵ E(K), f(e) ∉ E(K), assuming f(e) ≠ e. Several extensions and variations of this problem are also discussed here.