We study encodings from CSP into asynchronous CCS with name passing and matching, so in fact, the asynchronous π-calculus. By doing so, we discuss two different ways to map the multi-way synchronisation mechanism of CSP into the two-way synchronisation mechanism of CCS. Both encodings satisfy the criteria of Gorla except for compositionality, as both use an additional top-level context. Following the work of Parrow and Sjödin, the first encoding uses a centralised coordinator and establishes a variant of weak bisimilarity between source terms and their translations. The second encoding is decentralised, and thus more efficient, but ensures only a form of coupled similarity between source terms and their translations.

A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour.In this paper we obtain the first results on perfect matching width since its introduction. For the restricted case of bipartite graphs, we show that perfect matching width is equivalent to directed treewidth and thus the Directed Grid Theorem by Kawarabayashi and Kreutzer for directed treewidth implies Norine's conjecture.

We prove a recent conjecture of Beisegel et al. that for every positive integer $k$, every graph containing an induced $P_k$ also contains an avoidable $P_k$. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for $k \in \{1,2\}$ (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvátal et al. 2002, which assumed cycle restrictions.  We provide a constructive and elementary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.

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Tuza famously conjectured in 1981 that in a graph without k + 1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colourable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.