114 research outputs found
On Colorings of Graph Powers
In this paper, some results concerning the colorings of graph powers are
presented. The notion of helical graphs is introduced. We show that such graphs
are hom-universal with respect to high odd-girth graphs whose st power
is bounded by a Kneser graph. Also, we consider the problem of existence of
homomorphism to odd cycles. We prove that such homomorphism to a -cycle
exists if and only if the chromatic number of the st power of
is less than or equal to 3, where is the 2-subdivision of . We also
consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the
existence of high girth cubic graphs which are not homomorphic to the cycle of
size five. Several problems which are closely related to Ne\v{s}et\v{r}il's
problem are introduced and their relations are presented
Bounds for Visual Cryptography Schemes
In this paper, we investigate the best pixel expansion of the various models
of visual cryptography schemes. In this regard, we consider visual cryptography
schemes introduced by Tzeng and Hu [13]. In such a model, only minimal
qualified sets can recover the secret image and that the recovered secret image
can be darker or lighter than the background. Blundo et al. [4] introduced a
lower bound for the best pixel expansion of this scheme in terms of minimal
qualified sets. We present another lower bound for the best pixel expansion of
the scheme. As a corollary, we introduce a lower bound, based on an induced
matching of hypergraph of qualified sets, for the best pixel expansion of the
aforementioned model and the traditional model of visual cryptography realized
by basis matrices. Finally, we study access structures based on graphs and we
present an upper bound for the smallest pixel expansion in terms of strong
chromatic index
Note on the smallest root of the independence polynomial
One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
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