24 research outputs found
Determination of the prime bound of a graph
Given a graph , a subset of is a module of if for each
, is adjacent to all the elements of or to none
of them. For instance, , and () are
modules of called trivial. Given a graph , (respectively
) denotes the largest integer such that there is a module
of which is a clique (respectively a stable) set in with . A
graph is prime if and if all its modules are trivial. The
prime bound of is the smallest integer such that there is a prime
graph with , and . We establish the following. For every graph such that
and
is not an integer,
. Then, we prove that
for every graph such that where , or . Moreover if and only if or its complement
admits isolated vertices. Lastly, we show that for every non
prime graph such that and .Comment: arXiv admin note: text overlap with arXiv:1110.293
3-uniform hypergraphs: modular decomposition and realization by tournaments
Let be a 3-uniform hypergraph. A tournament defined on is
a realization of if the edges of are exactly the 3-element subsets of
that induce 3-cycles. We characterize the 3-uniform hypergraphs that
admit realizations by using a suitable modular decomposition
The simplicity index of tournaments
An -tournament with vertex set is simple if there is no subset
of such that and for every , either or . The simplicity
index of an -tournament is the minimum number of arcs whose
reversal yields a non-simple tournament. M\"{u}ller and Pelant (1974) proved
that , and that equality holds if and only if is
doubly regular. As doubly regular tournaments exist only if , for . In this paper, we
study the class of -tournaments with maximal simplicity index for