Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.20575

Framing Lattices and Flow Polytopes

Matias von Bell·Cesar Ceballos
View original source ↗Read source document ↗Export metadata ↓

Abstract

Flow polytopes of acyclic oriented graphs arise naturally in combinatorial optimization, and the study of their volumes and triangulations has revealed intriguing connections across combinatorics, geometry, algebra, and representation theory. In this work, we introduce the framing lattice associated with a framed graph, whose Hasse diagram is dual to a framed triangulation of the corresponding flow polytope. Framing lattices are remarkable in that they provide a unifying framework encompassing many classical and well-studied lattice structures, including the Boolean lattice, the Tamari lattice, and the weak order on permutations. They further subsume a broad array of examples such as all type-A Cambrian lattices, the Grassmann and grid-Tamari lattices, the alt-$ν$-Tamari and cross-Tamari lattices, the permutree lattices, and the $τ$-tilting posets of certain gentle algebras. We show, among several foundational structural properties, that the framing lattice is a semidistributive, congruence uniform, and polygonal lattice, with its polygons consisting of squares, pentagons, and hexagons. We study its connections to noncrossing partitions via Reading's core label orders, simple representations of its join and meet irreducible elements, and several of its lattice congruences and quotients induced by a graph operation called an M-move.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Matias von Bell, Cesar Ceballos. 2026-05-12. Framing Lattices and Flow Polytopes. https://arxiv.org/abs/2512.20575

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗