38,146 research outputs found
The typical structure of maximal triangle-free graphs
Recently, settling a question of Erd\H{o}s, Balogh and
Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most
-vertex maximal triangle-free graphs, matching the previously known lower
bound. Here we characterize the typical structure of maximal triangle-free
graphs. We show that almost every maximal triangle-free graph admits a
vertex partition such that is a perfect matching and is an
independent set.
Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the
Erd\H{o}s-Simonovits stability theorem, and recent results of
Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure
of independent sets in hypergraphs. The proof also relies on a new bound on the
number of maximal independent sets in triangle-free graphs with many
vertex-disjoint 's, which is of independent interest.Comment: 17 page
Three remarks on graphs
Let . A graph is if for any pairwise
disjoint independent vertex subsets in , there exist
pairwise disjoint maximum independent sets in such that
for . Recognizing graphs is
co-NP-hard, as shown by Chv\'atal and Hartnell (1993) and, independently, by
Sankaranarayana and Stewart (1992). Extending this result and answering a
recent question of Levit and Tankus, we show that recognizing
graphs is co-NP-hard for . On the positive side, we show that
recognizing graphs is, for each , FPT parameterized by
clique-width and by tree-width. Finally, we construct graphs that are not
such that, for every vertex in and every maximal
independent set in , the largest independent set in consists of a single vertex, thereby refuting a conjecture of
Levit and Tankus.Comment: 6 page
Vertex decomposability and regularity of very well-covered graphs
A graph is well-covered if it has no isolated vertices and all the
maximal independent sets have the same cardinality. If furthermore two times
this cardinality is equal to , the graph is called very
well-covered. The class of very well-covered graphs contains bipartite
well-covered graphs. Recently in \cite{CRT} it is shown that a very
well-covered graph is Cohen-Macaulay if and only if it is pure shellable.
In this article we improve this result by showing that is Cohen-Macaulay if
and only if it is pure vertex decomposable. In addition, if denotes the
edge ideal of , we show that the Castelnuovo-Mumford regularity of
is equal to the maximum number of pairwise 3-disjoint edges of . This
improves Kummini's result on unmixed bipartite graphs.Comment: 11 page
Maximal independent sets and separating covers
In 1973, Katona raised the problem of determining the maximum number of
subsets in a separating cover on n elements. The answer to Katona's question
turns out to be the inverse to the answer to a much simpler question: what is
the largest integer which is the product of positive integers with sum n? We
give a combinatorial explanation for this relationship, via Moon and Moser's
answer to a question of Erdos: how many maximal independent sets can a graph on
n vertices have? We conclude by showing how Moon and Moser's solution also
sheds light on a problem of Mahler and Popken's about the complexity of
integers.Comment: To appear in the Monthl
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
On the order of countable graphs
A set of graphs is said to be independent if there is no homomorphism between
distinct graphs from the set. We consider the existence problems related to the
independent sets of countable graphs. While the maximal size of an independent
set of countable graphs is 2^omega the On Line problem of extending an
independent set to a larger independent set is much harder. We prove here that
singletons can be extended (``partnership theorem''). While this is the best
possible in general, we give structural conditions which guarantee independent
extensions of larger independent sets. This is related to universal graphs,
rigid graphs and to the density problem for countable graphs
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