29 research outputs found
Homomorphisms and polynomial invariants of graphs
This paper initiates a general study of the connection between graph homomorphisms and the Tutte
polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte
polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials.
As an application, we describe in terms of homomorphism counting some fundamental evaluations of the
Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a
homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de AndalucÃa PAI-FQM-0164Junta de AndalucÃa P06-FQM-0164
Tutte uniqueness of locally grid graphs
A graph is said to be locally grid if the structure around each of
its vertices is a 3 × 3 grid. As a follow up of the research initiated
in [4] and [3] we prove that most locally grid graphs are uniquely
determined by their Tutte polynomial.Ministerio de Ciencia y TecnologÃa BFM2001-2474-ORIJunta de AndalucÃa PAI FQM-16
Locally grid graphs: classification and Tutte uniqueness
We define a locally grid graph as a graph in which the structure around each vertex is a 3×3 grid ⊞, the canonical examples being the toroidal grids Cp×Cq. The paper contains two main results. First, we give a complete classification of locally grid graphs, showing that each of them has a natural embedding in the torus or in the Klein bottle. Secondly, as a continuation of the research initiated in (On graphs determined by their Tutte polynomials, Graphs Combin., to appear), we prove that Cp×Cq is uniquely determined by its Tutte polynomial, for p,q⩾6
On the Ramsey numbers for stars versus complete graphs
For graphs G1, . . . , Gs, the multicolor Ramsey number R(G1, . . . , Gs) is the smallest integer r such that if we
give any edge col-oring of the complete graph on r vertices with s colors then there exists a monochromatic
copy of Gi colored with color i, for some 1 ≤ i ≤ s. In this work the multicolor Ramsey number
R(Kp1
, . . . , Kpm
, K1,q1
, . . . , K1,qn
) is determined for any set of com-plete graphs and stars in terms of R(Kp1
, . . . ,
Kpm
)Ministerio de Educación y Ciencia MTM2008-06620-C03-02Junta de AndalucÃa P06-FQM-0164
Exact value of 3 color weak Rado number
For integers k, n, c with k, n ≥ 1 and c ≥ 0, the n color weak Rado number
W Rk(n, c) is defined as the least integer N, if it exists, such that for every n coloring of the set {1, 2, ..., N}, there exists a monochromatic solution in that set
to the equation x1 + x2 + ... + xk + c = xk+1, such that xi = xj when i = j. If no
such N exists, then W Rk(n, c) is defined as infinite.
In this work, we consider the main issue regarding the 3 color weak Rado number
for the equation x1 + x2 + c = x3 and the exact value of the W R2(3, c) = 13c + 22
is established
3-color Schur numbers
Let k ≥ 3 be an integer, the Schur number Sk(3) is the least positive integer, such that for
every 3-coloring of the integer interval [1, Sk(3)] there exists a monochromatic solution to
the equation x1+ · · · + xk= xk+1, where xi
, i = 1, . . . , k need not be distinct.
In 1966, a lower bound of Sk(3) was established by Znám (1966). In this paper, we
determine the exact formula of Sk(3) = k
3 + 2k
2 − 2, finding an upper bound which
coincides with the lower bound given by Znám (1966). This is shown in two different
ways: in the first instance, by the exhaustive development of all possible cases and in the
second instance translating the problem into a Boolean satisfiability problem, which can
be handled by a SAT solver
Computing the Tutte polynomial of Archimedean tilings
We describe an algorithm to compute the Tutte polynomial of large fragments of Archimedean tilings by squares, triangles, hexagons and combinations thereof. Our algorithm improves a well known method for computing the Tutte polynomial of square lattices. We also address the problem of obtaining Tutte polynomial evaluations from the symbolic expressions generated by our algorithm, improving the best known lower bound for the asymptotics of the number of spanning forests, and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice
Equation-regular sets and the Fox–Kleitman conjecture
Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero
integer b such that the 2k-variable linear Diophantine equation
∑k
i=1
(xi − yi) = b
is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all
b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for
all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of
the corresponding equation for all b ≥ 1. In particular, this independently confirms the
conjecture for k = 3. We also briefly discuss the case k = 4
On the degree of regularity of a certain quadratic Diophantine equation
We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic
Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic
progressions