The main theorem gives the asymptotic sampling distribution and enumeration formulae for configurations, and accommodates forbidden edges, and enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.
Abstract
We develop a unified framework for constructing combinatorial structures under local constraints. Our approach extends the configuration model for random graphs with a prescribed degree sequence, and covers many special cases, including bipartite graphs, directed graphs, oriented graphs, edge-colored (bipartite) graphs, and (directed) hypergraphs. By reformulating half-edge matching as an independent set problem in an auxiliary graph, we identify 2-uniformity, a property characterising when greedy sampling preserves asymptotic uniformity. We classify all 2-uniform graphs and show that only two classes, the configuration space and the bipartite configuration space, have unbounded independence number, enabling the asymptotic regime. Our main theorem then gives the asymptotic sampling distribution and enumeration formulae for configurations, with error terms of order $O(d_{\max}^4\log m/m+d_{\max}^2(\log m)^2/m)$ as the number of edges $m$ tends to infinity with maximum degree $d_{\max}=O(m^{1/4}/\log m)$. This settles the long-standing $O(m^{1/4-\tau})$ bound (for some fixed $\tau>0$), making the critical exponent explicit. Furthermore, our theorem accommodates forbidden edges, provided that each vertex participates in at most $O(m^{1/4}/\log m)$ of them. In particular, this enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.
We investigate metric dimension and the localization game for several families of directed analogues of strongly regular graphs and their generalizations, adapting a probabilistic method of Babai (1980) for bounding the size of resolving sets in undirected strongly regular graphs. We derive upper bounds on the localization number and metric dimension depending on the order of the graph and the maximum number of common out-neighbours for a pair of vertices. We consider normally regular digraphs, so-called"ordinary graphs", classes of Deza digraphs, divisible design digraphs, nearly doubly regular tournaments, and certain doubly regular team tournaments. In particular, for asymmetric normally regular digraphs on $n$ vertices, we show that these invariants are bounded above by $O(\sqrt{n} \log n)$, and improve this to $O(\log n)$ for a class of doubly regular team tournaments.
This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincar\'e-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.
Matthias Hofmann, James B Kennedy, Delio Mugnolo et al.· 0 citations
We investigate the problem of computing the distribution function for the shortest and longest path lengths in a directed graph with random edge lengths. Specifically, when these lengths are uniformly distributed, the problem reduces to computing the volume of a polytope defined by the graph structure. We establish that the problem is $\#P$-hard, even under the restricted condition that the random edge lengths are identically and independently distributed (i.i.d.) according to any continuous probability distribution with certain natural conditions, the local uniformity. This hardness result applies broadly: while the uniform distribution provides an essential case for the reduction, other distributions -- such as exponential or normal -- are similarly hard because they contain uniform distributions in every arbitrarily small interval. Furthermore, we show that the problem is contained within $\mathrm{XP}$ with respect to the treewidth $k$ of the underlying undirected graph. For the specific case of i.i.d. uniform edge lengths, we present a novel dynamic programming algorithm that processes a tree decomposition by iteratively performing convolutions to propagate distribution functions. Our approach achieves a time complexity of $n^{O(k^2)}$ for any fixed treewidth $k$.
We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $\lambda$ is upper bounded by a constant and $\lambda(1 - \beta) \lesssim \Delta^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions.
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.
The RRS conjecture for constrained multi-relational graphons in the non-extremal regime is resolved, proving that entropy-maximizing solutions are step functions with finitely many blocks under the condition the subgraph density constraints are analytically independent and for almost all feasible combinations of sufficient statistics.