European Journal of Combinatorics最新文献

We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and 5. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new method for generating the combinatorial types of these polytopes via the classification of point set order types. In dimensions 4 and 5, there are 348 and 51 polytopes, respectively, yielding many new examples for further study (also discovered independently by Ma and Zheng).

We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $k\le 10$. It was shown by Vinberg in 1985 that for any $k$, we have $d\le 29$, and no better bounds have previously been published for $k\ge 5$. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.

Duchamp–Hivert–Thibon introduced the construction of a right $H_n\left(0\right)$-module, denoted as $M_P$, for any partial order $P$ on the set $\left[n\right]$. This module is defined by specifying a suitable action of $H_n\left(0\right)$ on the set of linear extensions of $P$. In this paper, we refer to this module as the poset module associated with $P$. Firstly, we show that ${⨁}_{n\ge 0}{G}_0\left(P\left(n\right)\right)$ has a Hopf algebra structure that is isomorphic to the Hopf algebra of quasisymmetric functions, where $P\left(n\right)$ is the full subcategory of $\mathrm{mod\; -}{H}_n\left(0\right)$ whose objects are direct sums of finitely many isomorphic copies of poset modules and $G_0\left(P\left(n\right)\right)$ is the Grothendieck group of $P\left(n\right)$. We also demonstrate how (anti-) automorphism twists interact with these modules, the induction product and restrictions. Secondly, we investigate the (type 1) quasisymmetric power sum expansion of some quasi-analogues $Y_{\alpha }$ of Schur functions, where $\alpha$ is a composition. We show that they can be expressed as the sum of the $P$-partition generating functions of specific posets, which allows us to utilize the result established by Liu–Weselcouch. Additionally, we provide a new algorithm for obtaining these posets. Using these findings, for the dual immaculate function and the extended Schur function, we express the coefficients appearing in the quasisymmetric power sum expansions in terms of border strip tableaux.

We investigate connected cubic vertex-transitive graphs whose edge sets admit a partition into a 2-factor $C$ and a 1-factor that is invariant under a vertex-transitive subgroup of the automorphism group of the graph and where the quotient graph with respect to $C$ is a cycle. There are two essentially different types of such cubic graphs. In this paper we focus on the examples of what we call the alternating type. We classify all such examples admitting a vertex-transitive subgroup of the automorphism group of the graph preserving the corresponding 2-factor and also determine the ones for which the 2-factor is invariant under the full automorphism group of the graph. In this way we introduce a new infinite family of cubic vertex-transitive graphs that is a natural generalization of the well-known generalized Petersen graphs as well as of the honeycomb toroidal graphs. The family contains an infinite subfamily of arc-regular examples and an infinite subfamily of 2-arc-regular examples.

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by ${\hat{R}}_r\left(H\right)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic copy of $H$. In the case of 2-uniform paths $P_n$, it is known that $\Omega \left(r^{2}n\right)={\hat{R}}_r\left(P_n\right)=O\left((r^{2}logr)n\right)$ with the best bounds essentially due to Krivelevich (2019). In a recent breakthrough result, Letzter et al. (2021) gave a linear upper bound on the $r$-color size-Ramsey number of the $k$-uniform tight path $P_n^{\left(k\right)}$; i.e. ${\hat{R}}_r\left(P_n^{\left(k\right)}\right)=O_{r,k}\left(n\right)$. At about the same time, Winter (2023) gave the first non-trivial lower bounds on the 2-color size-Ramsey number of $P_n^{\left(k\right)}$ for $k\ge 3$; i.e.