Oriol Serra

Efficient dominating sets in Cayley graphs

An independent set C of vertices in a graph is an efficient dominating set (or perfect code) when each vertex not in C is adjacent to exactly one vertex in C. An E-chain is a countable family of nested graphs, each of which has an efficient dominating set. The Hamming codes in the n-cubes provide a classical example of E-chains. We give a constructing tool to produce E-chains of Cayley graphs. This tool is used to construct infinite families of E-chains of Cayley graphs on symmetric groups. These families include the well-known star graphs, for which the efficient domination property was proved by Arumugam and Kala, and pancake graphs. Additional structural properties of the E-chains and the efficient dominating sets involved are also presented. Given a tree T, the T-graph associated to T seems to be a natural candidate of a graph with an efficient dominating set. However, we prove that a T-graph has an efficient dominating set if and only if T is a star.

A combinatorial proof of the Removal Lemma for Groups

Green [B. Green, A Szemeredi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemeredi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

Transversals of additive Latin squares

AbstractLetA={a1, …,ak} andB={b1, …,bk} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈Sksuch that the sums αi+bi, 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤp)a and

On sums of dilates

\mathbb{Z}_{p^a }

On large (D, D )-graphs

in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order.

On Monochromatic solutions of equations in groups

A digraphX is said to be Vosperian if any fragment has cardinality either 1 or|V(X)| − d+(X) − 1.A digraph is said to be superconnected if every minimum cutset is the set of vertices adjacent from or to some vertex.In this paper we characterize Vosperian and superconnected Abelian Cayley directed graphs. Our main tool is a difficult theorem of J.H. Kemperman from Additive Group Theory.In particular we characterize Vosperian and superconnected loops network (also called circulants).

Cayley Digraphs Based on the de Bruijn Networks

For k prime and A a finite set of integers with |A| ≥ 3(k − 1)2(k − 1)! we prove that |A + k · A| ≥ (k + 1)|A| − ⌈k(k + 2)/4⌉ where k · A = {ka: a ∈ A}. We also describe the sets for which equality holds.

Construction of k -arc transitive digraphs

G be a nonabelian torsion-free group. Let C be a finite generating subset of G such that . We prove that, for all subsets B of G with , we have .In particular, a finite subset X with cardinality satisfies the inequality if and only if there are elements , such that the following two conditions hold:(i) .(ii) where .