Julián Salas

Improving the characterization of P-stability for applications in network privacy

Recently, we have found that the concept of P-stability has interesting applications in network privacy. In the context of Online Social Networks it may be used for obtaining a fully polynomial randomized approximation scheme for graph masking and measuring disclosure risk. Also by using the characterization for P-stable sequences from Jerrum, McKay and Sinclair (1992) it is possible to obtain optimal approximations for the problem of k -degree anonymity. In this paper, we present results on P-stability considering the additional restriction that the degree sequence must not intersect the edges of an excluded graph X , improving earlier results on P-stability. As a consequence we extend the P-stable classes of scale-free networks from Torra et?al. (2015), obtain an optimal solution for k -anonymity and prove that all the known conditions for P-stability are sufficient for sequences to be graphic.

On the order of graphs with a given girth pair

A (k;g,h)-graph is a k-regular graph of girth pair (g,h) where g is the girth of the graph, h is the length of a smallest cycle of different parity than g and g

A new bound for the connectivity of cages

Abstract An ( r , g ) -cage is an r -regular graph of girth g of minimum order. We prove that all ( r , g ) -cages are at least ⌈ r / 2 ⌉ -connected for every odd girth g ≥ 7 by means of a matrix technique which allows us to construct graphs without short cycles. This lower bound on the vertex connectivity of cages is a new advance in proving the conjecture of Fu, Huang and Rodger which states that all ( r , g ) -cages are r -connected.

On a conjecture on the order of cages with a given girth pair

A ( k ; g , h ) -graph is a k -regular graph of girth pair ( g , h ) where g is the girth of the graph, h is the length of a smallest cycle of different parity than g and g < h . A ( k ; g , h ) -cage is a ( k ; g , h ) -graph with the least possible number of vertices denoted n ( k ; g , h ) . Harary and Kovacs (1983) conjectured the inequality n ( k ; g , h ) � n ( k , h ) for all k � 3 , g � 3 , h � g + 1 . In this paper, we prove this conjecture for all ( k ; g , h ) -cage with g odd provided that a bipartite ( k , h ) -cage exists. When g is even we prove the conjecture for h � 2 g - 1 , provided that a bipartite ( k , g ) -cage exists.

The degree/diameter problem in maximal planar bipartite graphs

The (?;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ? and diameter D. We consider the (?;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (?; 2) problem, the number of vertices is n = ?+2; and for the (?; 3) problem, n = 3??1 if ? is odd and n = 3? ? 2 if ? is even. Then, we study the general case (?;D) and obtain that an upper bound on n is approximately 3(2D + 1)(? ? 2)?D=2? and another one is C(? ? 2)?D=2? if ? D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (? ? 2)k if D = 2k, and 3(? ? 3)k if D = 2k + 1, for ? and D sufficiently large in both cases.

Approximating degree sequences with regular graphic sequences (Extended Abstract)

Abstract We study conditions to approximate the degree sequence of a given graph by a regular one. We obtain optimal conditions for a few metrics such as the edge rotation distance for graphs, the rectilinear and Euclidean distances over degree sequences. When we require the approximation to have at least k copies of each value in the degree sequence, that is, each value d in the degree sequence appears at least k times, the problem has a direct application in the context of data privacy when k-anonymity is required.

Families of small regular graphs of girth 7

The first known families of cages arised from the incidence graphs of generalized polygons of order

Edge-superconnectivity of semiregular cages with odd girth

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New results on connectivity of cages

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On the λ′-optimality in graphs with odd girth g and even girth h

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