Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization. Claude Berge made a conjecture about them, that was proved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and is now called the strong perfect graph theorem. This is a survey about perfect graphs, mostly focused on the strong perfect gr...

We construct highly edge-connected $r$-regular graph which do not contain $r-2$ pairwise disjoint perfect matchings. The results partially answer a question stated by Thomassen [Factorizing regular graphs, J. Comb. Theory Ser. B (2019), https://doi.org/10.1016/j.jctb.2019.05.002 (article in press)].

Perfect matchings and series-parallel graphs: multiplicatives proof nets as R&B-graphs Extended Abstract] Abstract A graph-theoretical look at multiplicative proof nets lead us to two new descriptions of a proof net, both as a graph endowed with a perfect matching. The rst one is a rather conventional encoding of the connectives which nevertheless allows us to unify various sequentialisation te...

An r-graph is an r-regular graph where every odd set of vertices is connected by at least r edges to the rest of the graph. Seymour conjectured that any r-graph is r + 1edge-colorable, and also that any r-graph contains 2r perfect matchings such that each edge belongs to two of them. We show that the minimum counter-example to either of these conjectures is a brick. Furthermore we disprove a va...

We prove that every perfect torsion theory for a ring R is differential (in the sense of [2]). In this case, we construct the extension of a derivation of a right R-module M to a derivation of the module of quotients of M . Then, we prove that the Lambek and Goldie torsion theories for any R are differential.

Let (R,m, k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dimR = d. Let M be a finitely generated R-module. We show that there exists β(M) ∈ R such that λ(M/I M) = eHK(M)q d + β(M)q + O(q).

A set of vertices in a graph is perfect dominating if every vertex outside the set is adjacent to exactly one vertex in the set, and is neighborhood connected if the subgraph induced by its open neighborhood is connected. In any graph the full set of vertices is perfect dominating, and in every connected graph the full set of vertices is neighborhood connected. It is shown that (i) in a connect...