4,067 research outputs found
Cartesian product of hypergraphs: properties and algorithms
Cartesian products of graphs have been studied extensively since the 1960s.
They make it possible to decrease the algorithmic complexity of problems by
using the factorization of the product. Hypergraphs were introduced as a
generalization of graphs and the definition of Cartesian products extends
naturally to them. In this paper, we give new properties and algorithms
concerning coloring aspects of Cartesian products of hypergraphs. We also
extend a classical prime factorization algorithm initially designed for graphs
to connected conformal hypergraphs using 2-sections of hypergraphs
Packing chromatic vertex-critical graphs
The packing chromatic number of a graph is the smallest
integer such that the vertex set of can be partitioned into sets ,
, where vertices in are pairwise at distance at least .
Packing chromatic vertex-critical graphs, -critical for short, are
introduced as the graphs for which
holds for every vertex of . If , then is
--critical. It is shown that if is -critical,
then the set can be almost
arbitrary. The --critical graphs are characterized, and
--critical graphs are characterized in the case when they
contain a cycle of length at least which is not congruent to modulo
. It is shown that for every integer there exists a
--critical tree and that a --critical
caterpillar exists if and only if . Cartesian products are also
considered and in particular it is proved that if and are
vertex-transitive graphs and , then is -critical
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