M.P. Revuelta

Weak Schur numbers and the search for G.W. Walker's lost partitions

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 a partition of {1,...,196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k=6, we obtain WS(6)>=572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver.

On the degree of regularity of a certain quadratic Diophantine equation

Abstract We show that, for every positive integer r , there exists an integer b = b ( r ) such that the 4-variable quadratic Diophantine equation ( x 1 − y 1 ) ( x 2 − y 2 ) = b is r -regular. Our proof uses Szemeredi’s theorem on arithmetic progressions.

Equation-regular sets and the Fox–Kleitman conjecture

Abstract Given k ≥ 1 , the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2 k -variable linear Diophantine equation ∑ i = 1 k ( x i − y i ) = b is ( 2 k − 1 ) -regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1 , this equation is not 2 k -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 finiteness of some n-color Rado numbers

Abstract For integers k , n , c with k , n ≥ 1 , the n -color Rado number R k ( n , c ) is defined to be the least integer N if any, or infinity otherwise, such that for every n -coloring of the set { 1 , 2 , … , N } , there exists a monochromatic solution in that set to the linear equation x 1 + x 2 + ⋯ + x k + c = x k + 1 . A recent conjecture of ours states that R k ( n , c ) should be finite if and only if every divisor d ≤ n of k − 1 also divides c . In this paper, we complete the verification of this conjecture for all k ≤ 7 . As a key tool, we first prove a general result concerning the degree of regularity over subsets of Z of some linear Diophantine equations.

Homomorphisms and Polynomial Invariants of Graphs

Abstract 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 Groups and Statistical Physics. A new type of uniqueness of graphs, strongly related to chromatically-unique graphs and Tutte-unique graphs, is introduced in order to provide a new point of view of the conjectures about uniqueness of graphs stated by Bollobas, Peabody and Riordan.

The dimension of a graph

Abstract 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 forbidden graphs for dimension 2 is also shown.

A general lower bound on the weak Schur number

Abstract For integers k, n with k , n ≥ 1 , the n-color weak Schur number W S k ( 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 x 1 , … , x k , x k + 1 in that interval to the equation x 1 + x 2 + … + x k = x k + 1 , with x i ≠ x j , when i ≠ j . We show a relationship between W S k ( n + 1 ) and W S k ( n ) and a general lower bound on the W S k ( n ) is obtained.

On the n-Color Weak Rado Numbers for the Equation

ABSTRACT For integers k, n, c with k, n ⩾ 1, and c ⩾ 0, the n-color weak Rado number is defined as the least integer N, if it exists, 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 with xi ≠ xj, when i ≠ j. If no such N exists, then is defined as infinite. In this paper, we determine the exact value of some of these numbers for n = 2 and n = 3, namely , for all c ⩾ 0 and for all c > 0. Our method consists in translating the problem into a Boolean satisfiability problem, which can then be handled by a SAT solver or by backtrack programming in the language C.

Exact value of 3 color weak Rado number

Abstract For integers k, n, c with k, n ≥ 1 and c ≥ 0 , the n color weak Rado number W R k ( 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 x 1 + x 2 + … + x k + c = x k + 1 , such that x i ≠ x j when i ≠ j . If no such N exists, then W R k ( n , c ) is defined as infinite. In this work, we consider the main issue regarding the 3 color weak Rado number for the equation x 1 + x 2 + c = x 3 and the exact value of the W R 2 ( 3 , c ) = 13 c + 22 is established.

Computing the Tutte polynomial of Archimedean tilings

Abstract 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.