16,521 research outputs found
Identifying codes of corona product graphs
For a vertex of a graph , let be the set of with all of
its neighbors in . A set of vertices is an {\em identifying code} of
if the sets are nonempty and distinct for all vertices . If
admits an identifying code, we say that is identifiable and denote by
the minimum cardinality of an identifying code of . In this
paper, we study the identifying code of the corona product of graphs
and . We first give a necessary and sufficient condition for the
identifiable corona product , and then express in terms of and the (total) domination number of .
Finally, we compute for some special graphs
Shellable graphs and sequentially Cohen-Macaulay bipartite graphs
Associated to a simple undirected graph G is a simplicial complex whose faces
correspond to the independent sets of G. We call a graph G shellable if this
simplicial complex is a shellable simplicial complex in the non-pure sense of
Bjorner-Wachs. We are then interested in determining what families of graphs
have the property that G is shellable. We show that all chordal graphs are
shellable. Furthermore, we classify all the shellable bipartite graphs; they
are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give an
recursive procedure to verify if a bipartite graph is shellable. Because
shellable implies that the associated Stanley-Reisner ring is sequentially
Cohen-Macaulay, our results complement and extend recent work on the problem of
determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We
also give a new proof for a result of Faridi on the sequentially
Cohen-Macaulayness of simplicial forests.Comment: 16 pages; more detail added to some proofs; Corollary 2.10 was been
clarified; the beginning of Section 4 has been rewritten; references updated;
to appear in J. Combin. Theory, Ser.
The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles
We prove that, for the edge ideal of a graph whose cycles are pairwise
vertex-disjoint, the arithmetical rank is bounded above by the sum of the
number of cycles and the maximum height of its associated primes
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