REVUELTA MARCHENA , María Pastora

El matemático alemán Euler (1707-1782) resolvió en 1736 el famoso problema de los puentes de Königsberg. Dicha ciudad estaba divida en cuatro partes, conectadas por siete puentes, al pasar por ella un río (ver Figura 1.1). El problema planteado era el siguiente: empezando a andar en un punto cualquiera de la ciudad ¿es posible volver al punto...

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x+y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) ≤ Rn(3)−2, between the Schur number S(n) and the Ramsey number Rn(3) = R(3, . . . , 3),...

For each graph G the dimension of G is defined as the smallest dimension in the Euclidean Space where there is an embedding in which all the edges of G are segments of a straight line of length one. The exact value is calculated for some important families of graphs and this value is compared with other invariants. An infinite quantity of...

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For any positive integers l and m, a set of integers is said to be (weakly) l-sum free modulo m if it contains no (pairwise distinct) elements x1, x2, . . . , xl , y satisfying the congruence x1 + . . . + xl ≡ y mod m. It is proved that, for any positive integers k and l, there exists a largest integer n for which the set of the first n...

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...

This paper initiates a study of the connection between graph homomorphisms and the Tutte polynomial. This connection enables us to extend the study to other important polynomial invariants associated with graphs, and closely related to the Tutte polynomial. We then obtain applications of these relationships in several areas, including Abelian...

For integers k, n with k, n ≥ 1, the n-color weak Schur number W Sk(n) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1, . . . , xk, xk+1 in that interval to the equation x1 +x2 +. . .+xk = xk+1, with xi 6= xj , when i 6= j. We show a relationship...

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.

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.

F whic or h ev is ery p balanced, ositive in i.e., teger whose n ≡ 0 asso mod ciated 4, we Steinhaus construct a triangle zero-sum contains {±1} as man -sequence y +1's of as length 1's. n This implies the existence of balanced binary sequences of every length m 0 or 3 mo − d 4, ≡ thereby providing a new solution to a problem posed by Steinhaus in 1963.

In this study, we focus on the concept of the 2-color off-diagonal generalized weak Schur numbers, denoted as WS(2; k1, k2). These numbers are defined for integers ki ≥ 2, where i = 1, 2, as the smallest integer M, such that any 2-coloring of the integer interval [1, M] must contain a 2-colored solution to the equation Ekj: x1 + x2 + ... + xkj...

For integers k, n with k, n ≥ 1, the n-color weak Schur number WSk (n) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1,..., xk , xk+1 in that interval to the equation: x1 + x2 +···+ xk = xk+1, with xi = x j , when i = j. In this paper, we obtain the exact...

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....

In this paper we compute the Frobenius number of certain Fibonacci numerical semigroups, that is, numerical semigroups generated by a set of Fibonacci numbers, in terms of Fibonacci numbers.

We give a complete classification of hexagonal tilings and locally C6 graphs, by showing that each of them has a natural embedding in the torus or in the Klein bottle (see [12]). We also show that locally grid graphs, defined in [9, 12], are minors of hexagonal tilings (and by duality of locally C6 graphs) by contraction of a particular perfect...

A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof. Here we show WS(5) ≥ 196, by constructing...

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 ,...

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,...

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...