Ran Ziv

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets Vi of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each Vi at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s.

A Tree Version of Kőnig's Theorem

Kőnigs theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Halls theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed.

Note: A note on the edge cover number and independence number in hypergraphs

Consider a hypergraph of rank r>2 with m edges, independence number @a and edge cover number @r. We prove the inequality@r=<(r-2)m+@ar-1.One application of this inequality is a special case of a conjecture of Aharoni and the first author extending Rysers Conjecture to matroids.

On a conjecture of Stein

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of

Fair Representation by Independent Sets

K_{n,n}

Points with large α-depth

Kn,n is partitioned into n sets, each of size n, then there is a partial rainbow matching of size

Uniqueness of the extreme cases in theorems of Drisko and Erdős–Ginzburg–Ziv

n-1