Home » Author index

Aharoni, Ron


Graph Theory » Infinite Graphs

Strong matchings and covers ★★★

Author(s): Aharoni

Let $ H $ be a hypergraph. A strongly maximal matching is a matching $ F \subseteq E(H) $ so that $ |F' \setminus F| \le |F \setminus F'| $ for every matching $ F' $. A strongly minimal cover is a (vertex) cover $ X \subseteq V(H) $ so that $ |X' \setminus X| \ge |X \setminus X'| $ for every cover $ X' $.

Conjecture   If $ H $ is a (possibly infinite) hypergraph in which all edges have size $ \le k $ for some integer $ k $, then $ H $ has a strongly maximal matching and a strongly minimal cover.

Keywords: cover; infinite graph; matching

Posted by mdevos
updated October 23rd, 2007
add new comment
Combinatorics » Matroid Theory

Aharoni-Berger conjecture ★★★

Author(s): Aharoni; Berger

Conjecture   If $ M_1,\ldots,M_k $ are matroids on $ E $ and $ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $ for every partition $ \{X_1,\ldots,X_k\} $ of $ E $, then there exists $ X \subseteq E $ with $ |X| = \ell $ which is independent in every $ M_i $.

Keywords: independent set; matroid; partition

Posted by mdevos
updated June 11th, 2007
add new comment
Graph Theory » Coloring » Vertex coloring

Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let $ r $ be a positive integer. We say that a graph $ G $ is strongly $ r $-colorable if for every partition of the vertices to sets of size at most $ r $ there is a proper $ r $-coloring of $ G $ in which the vertices in each set of the partition have distinct colors.

Conjecture   If $ \Delta $ is the maximal degree of a graph $ G $, then $ G $ is strongly $ 2 \Delta $-colorable.

Keywords: strong coloring

Posted by berger
updated November 19th, 2009
5 comments
Syndicate content