422 research outputs found
Threshold Graphs Maximize Homomorphism Densities
Given a fixed graph and a constant , we can ask what graphs
with edge density asymptotically maximize the homomorphism density of
in . For all for which this problem has been solved, the maximum is
always asymptotically attained on one of two kinds of graphs: the quasi-star or
the quasi-clique. We show that for any the maximizing is asymptotically
a threshold graph, while the quasi-clique and the quasi-star are the simplest
threshold graphs having only two parts. This result gives us a unified
framework to derive a number of results on graph homomorphism maximization,
some of which were also found quite recently and independently using several
different approaches. We show that there exist graphs and densities
such that the optimizing graph is neither the quasi-star nor the
quasi-clique, reproving a result of Day and Sarkar. We rederive a result of
Janson et al. on maximizing homomorphism numbers, which was originally found
using entropy methods. We also show that for large enough all graphs
maximize on the quasi-clique, which was also recently proven by Gerbner et al.,
and in analogy with Kopparty and Rossman we define the homomorphism density
domination exponent of two graphs, and find it for any and an edge
On the Sandpile group of the cone of a graph
In this article, we give a partial description of the sandpile group of the
cone of the cartesian product of graphs in function of the sandpile group of
the cone of their factors. Also, we introduce the concept of uniform
homomorphism of graphs and prove that every surjective uniform homomorphism of
graphs induces an injective homomorphism between their sandpile groups. As an
application of these result we obtain an explicit description of a set of
generators of the sandpile group of the cone of the hypercube of dimension d.Comment: 20 pages, 11 figures. The title was changed, other impruvements were
made throughout the article. To appear in Linear Algebra and Its Application
Embedding into bipartite graphs
The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher,
Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any
, every balanced bipartite graph on vertices with bounded degree
and sublinear bandwidth appears as a subgraph of any -vertex graph with
minimum degree , provided that is sufficiently large. We show
that this threshold can be cut in half to an essentially best-possible minimum
degree of when we have the additional structural
information of the host graph being balanced bipartite. This complements
results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and
Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding
minimum degree threshold for -factors, with and fixed.
Moreover, it implies that the set of Hamilton cycles of is a generating
system for its cycle space.Comment: 16 pages, 2 figure
On the decomposition threshold of a given graph
We study the -decomposition threshold for a given graph .
Here an -decomposition of a graph is a collection of edge-disjoint
copies of in which together cover every edge of . (Such an
-decomposition can only exist if is -divisible, i.e. if and each vertex degree of can be expressed as a linear combination of
the vertex degrees of .)
The -decomposition threshold is the smallest value ensuring
that an -divisible graph on vertices with
has an -decomposition. Our main results imply
the following for a given graph , where is the fractional
version of and :
(i) ;
(ii) if , then
;
(iii) we determine if is bipartite.
In particular, (i) implies that . Our proof
involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory,
Series
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
H-colorings of large degree graphs
We consider the H-coloring problem on graphs with
vertices of large degree. We prove that for H an odd cycle,
the problem belongs to P. We also study the phase transition
of the problem, for an infinite family of graphs of a given
chromatic number, i.e. the threshold density value for which
the problem changes from P to NP-complete. We extend the result
for the case that the input graph has a logarithmic size of
small degree vertices.As a corollary, we get a new result on
the chromatic number; a new family of graphs, for which computing
the chromatic number can be done in polynomial time.Postprint (published version
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
A note on "Folding wheels and fans."
In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and
Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique
numbers onto which wheels and fans fold. We present an interpolation theorem
which generalizes their theorems 4.2 and 5.2. We show that their formula for
wheels is wrong. We show that for threshold graphs, the achromatic number and
folding number coincides with the chromatic number
- …