Luis Boza

A new method for classifying complex filiform Lie algebras

In this paper, we describe a new method to classify complex filiform Lie algebras based on the concept of isomorphism between Lie algebras. This method, which has the advantage of being applied to any dimension, gives the families of algebras in each dimension in an explicit way. In order to apply, only the corresponding structure theorem of complex filiform Lie algebras in each dimension is needed. As a consequence of our study, we also predict that the increase (in terms of quotiens) in the number of algebras families when passing from even dimension to odd dimension tends to 1 whereas it grows in a no finite way if passing from odd dimension to immediate even dimension.

Complex filiform Lie algebras of dimension 11

In this paper we give the explicit classification of complex filiform Lie algebras of dimension 11. To do this, we use a method previously obtained by us in an earlier paper, which is based on the concept of isomorphism between Lie algebras. At present, this explicit classification is not known, although Gomez, Jimenez and Khakimdjanov gave a list of these algebras in a non-explicit way, but in terms of cocycles. We find that there exist 188 families of complex filiform Lie algebras of dimension 11.

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.

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.

Miscellaneous properties of embeddings of line, total and middle graphs

Abstract Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that R (G) is planar, where R is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not R (G) is planar in an O (n) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The first characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs.

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.