25 research outputs found
Hypomorphy of graphs up to complementation
Let be a set of cardinality (possibly infinite). Two graphs and
with vertex set are {\it isomorphic up to complementation} if is
isomorphic to or to the complement of . Let be a
non-negative integer, and are {\it -hypomorphic up to
complementation} if for every -element subset of , the induced
subgraphs and are isomorphic up to
complementation. A graph is {\it -reconstructible up to complementation}
if every graph which is -hypomorphic to up to complementation is in
fact isomorphic to up to complementation. We give a partial
characterisation of the set of pairs such that two graphs
and on the same set of vertices are equal up to complementation
whenever they are -hypomorphic up to complementation. We prove in particular
that contains all pairs such that . We
also prove that 4 is the least integer such that every graph having a
large number of vertices is -reconstructible up to complementation; this
answers a question raised by P. Ill
Reconstruction of a coloring from its homogeneous sets
We study a reconstruction problem for colorings. Given a finite or countable
set , a coloring on is a function , where
is the collection of all 2-elements subsets of . A set is homogeneous for when is constant on . Let
be the collection of all homogeneous sets for . The
coloring is called the complement of . We say that
is {\em reconstructible} up to complementation from its homogeneous
sets, if for any coloring on such that we
have that either or . We present several
conditions for reconstructibility and non reconstructibility. We show that
there is a Borel way to reconstruct a coloring from its homogeneous sets
Edge ideals: algebraic and combinatorial properties
Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on
the algebraic and combinatorial properties of R/I(C) and C, respectively. We
give a criterion to estimate the regularity of R/I(C) and apply this criterion
to give new proofs of some formulas for the regularity. If C is a clutter and
R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity
of the ideal of vertex covers of C and give a formula for the projective
dimension of R/I(C). We also examine the associated primes of powers of edge
ideals, and show that for a graph with a leaf, these sets form an ascending
chain
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM