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Distance-generalized Core Decomposition
The -core of a graph is defined as the maximal subgraph in which every
vertex is connected to at least other vertices within that subgraph. In
this work we introduce a distance-based generalization of the notion of
-core, which we refer to as the -core, i.e., the maximal subgraph in
which every vertex has at least other vertices at distance within
that subgraph. We study the properties of the -core showing that it
preserves many of the nice features of the classic core decomposition (e.g.,
its connection with the notion of distance-generalized chromatic number) and it
preserves its usefulness to speed-up or approximate distance-generalized
notions of dense structures, such as -club.
Computing the distance-generalized core decomposition over large networks is
intrinsically complex. However, by exploiting clever upper and lower bounds we
can partition the computation in a set of totally independent subcomputations,
opening the door to top-down exploration and to multithreading, and thus
achieving an efficient algorithm
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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