arXiv subjects

Alain Bretto

Publications and source records attributed to Alain Bretto.

7 recordsLinked to original sources

arXiv2026

Characteristic adjacency matrix associated with a hypergraph

We define a new matrix associated with a hypergraph. A study of this matrix is carried out, in particular the study of its spectrum, which shows the relevance of the construction. We demon- strate that this adjacency matrix is characterisitic of the hypergraph, that is to say, two isomorphic hypergraphs have similar matrices. Furthermore, starting from this matrix, we can reconstruct the hypergraph. Starting from this matrix, we introduce new graphs associated with the hypergraph.

math.CO↗

arXiv2026

Linear representation of groups associated to graphs

In this article, we introduce a group linear representation associated to a graph. We study this representation as well as the canonical decomposition of the associated module. A character study is conducted, demonstrating the relevance of this approach to graph theory. We also develop the links that exist between algebra and graphs, particularly in the language of representations.

math.RT↗

arXiv2024

On the strong persistence property and normally torsion-freeness of square-free monomial ideals

In this paper, we first show that any square-free monomial ideal in $K[x_1, x_2, x_3, x_4, x_5]$ has the strong persistence property. Next we will provide a criterion for a minimal counterexample to the Conforti-Cornuejols conjecture. Finally we give a necessary and sufficient condition to determine the normally torsion-freeness of a linear combination of two normally torsion-free square-free monomial ideals.

math.AC↗

arXiv2024

Cayley graphs and G-graphs of Gyro-groups

In this paper the structure of the Cayley graphs and G-graphs of some gyro-groups are studied and some properties of them will be proved. Moreover we review some special gyro-groups including: gyro-commutative gyrogroups, dihedral gyro-groups and dihedralized gyro-groups. Then we try to establish some important properties of their associated G-graphs. Finally we will examine the symmetry of Cayley graphs and G-graphs of some gyro-groups.

math.CO↗

arXiv2024

About Berge-F\"uredi's conjecture on the chromatic index of hypergraphs

We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-F\"uredi conjectured bound $\mathrm{q}(\mathcal{H})\le \Delta([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the maximum degree $\Delta(\mathcal{H})$. This provides sharp information in connection with Erd\H{o}s-Faber-Lov\'asz Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex.

math.CO↗

arXiv2024

The Berge-F\"uredi conjecture on the chromatic index of hypergraphs with large hyperedges

This paper is concerned with two conjectures which are intimately related. The first is a generalization to hypergraphs of Vizing's Theorem on the chromatic index of a graph and the second is the well-known conjecture of Erd\H{o}s, Faber and Lov\'asz which deals with the problem of coloring a family of cliques intersecting in at most one vertex. We are led to study a special class of uniform and linear hypergraphs for which a number of properties are established.

math.CO↗

arXiv2009

Cartesian product of hypergraphs: properties and algorithms

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspects of Cartesian products of hypergraphs. We also extend a classical prime factorization algorithm initially designed for graphs to connected conformal hypergraphs using 2-sections of hypergraphs.

cs.DM↗