414 research outputs found
Toric residue and combinatorial degree
Consider an n-dimensional projective toric variety X defined by a convex
lattice polytope P. David Cox introduced the toric residue map given by a
collection of n+1 divisors Z_0,...,Z_n on X. In the case when the Z_i are
T-invariant divisors whose sum is X\T the toric residue map is the
multiplication by an integer number. We show that this number is the degree of
a certain map from the boundary of the polytope P to the boundary of a simplex.
This degree can be computed combinatorially.
We also study radical monomial ideals I of the homogeneous coordinate ring of
X. We give a necessary and sufficient condition for a homogeneous polynomial of
semiample degree to belong to I in terms of geometry of toric varieties and
combinatorics of fans.
Both results have applications to the problem of constructing an element of
residue one for semiample degrees.Comment: 13 pages, one section added, 1 pstex figure. To appear in Trans.
Amer. Math. So
Interval non-edge-colorable bipartite graphs and multigraphs
An edge-coloring of a graph with colors is called an interval
-coloring if all colors are used, and the colors of edges incident to any
vertex of are distinct and form an interval of integers. In 1991 Erd\H{o}s
constructed a bipartite graph with 27 vertices and maximum degree 13 which has
no interval coloring. Erd\H{o}s's counterexample is the smallest (in a sense of
maximum degree) known bipartite graph which is not interval colorable. On the
other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree
at most 3 have an interval coloring. In this paper we give some methods for
constructing of interval non-edge-colorable bipartite graphs. In particular, by
these methods, we construct three bipartite graphs which have no interval
coloring, contain 20,19,21 vertices and have maximum degree 11,12,13,
respectively. This partially answers a question that arose in [T.R. Jensen, B.
Toft, Graph coloring problems, Wiley Interscience Series in Discrete
Mathematics and Optimization, 1995, p. 204]. We also consider similar problems
for bipartite multigraphs.Comment: 18 pages, 7 figure
On the number of k-dominating independent sets
We study the existence and the number of -dominating independent sets in
certain graph families. While the case namely the case of maximal
independent sets - which is originated from Erd\H{o}s and Moser - is widely
investigated, much less is known in general. In this paper we settle the
question for trees and prove that the maximum number of -dominating
independent sets in -vertex graphs is between and
if , moreover the maximum number of
-dominating independent sets in -vertex graphs is between
and . Graph constructions containing a large number of
-dominating independent sets are coming from product graphs, complete
bipartite graphs and with finite geometries. The product graph construction is
associated with the number of certain MDS codes.Comment: 13 page
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