85 research outputs found
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max
formula for the maximum cardinality of the union of k sink-stable sets. The
results imply a recent min-max theorem of Abeledo and Atkinson on the Clar
number of bipartite plane graphs and a sharpening of Minty's coloring theorem.
We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on
cyclic stable sets
Red-blue clique partitions and (1-1)-transversals
Motivated by the problem of Gallai on -transversals of -intervals,
it was proved by the authors in 1969 that if the edges of a complete graph
are colored with red and blue (both colors can appear on an edge) so that there
is no monochromatic induced and then the vertices of can be
partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani
recently strengthened this by showing that it is enough to assume that there is
no induced monochromatic and there is no induced in {\em one of the
colors}. Here this is strengthened further, it is enough to assume that there
is no monochromatic induced and there is no on which both color
classes induce a .
We also answer a question of Kaiser and Rabinovich, giving an example of six
-convex sets in the plane such that any three intersect but there is no
-transversal for them
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max formula for the maximum cardinality of the union of k sink-stable sets. The results imply a recent min-max theorem of Abeledo and Atkinson on the Clar number of bipartite plane graphs and a sharpening of Minty’s coloring theorem. We also exhibit a link to min-max results of Bessy and Thomasse ́ and of Sebő on cyclic stable sets
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