Discrete Mathematics Papers of the Week

A direct application of the symmetric Lovász Local Lemma shows that $\textrm{HG}(G) < e\Delta$ for every graph $G$ of maximum degree at most $\Delta \ge 1$. We prove that this bound is asymptotically sharp: for every $\varepsilon > 0$ and every sufficiently large $\Delta$, there exists a graph $G$ of maximum degree at most $\Delta$ with $\textrm{HG}(G) \ge (e - \varepsilon)\Delta$.

We determine it on $\ell_1$ products of betweenness-free metrics, those with no point between two others, a class containing every Hamming graph.

We determine the edge-structure and edge-substructure fault diameters of the exchanged hypercube $EH(s,t)$ for every $1\le s\le t$.

We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25].

In particular, we show that in monadically dependent classes, tuples of fixed length have only $|U|^{O(1)}$ types of fixed quantifier rank over any set of parameters $U$.

Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small.

We give a unified framework for one-sided testability (constant-query testing) of metric families.

We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top.

We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity.

Unlike conventional consensus, which requires the system state errors to converge exactly to the origin, practical consensus only requires the errors to converge to a bounded neighborhood of the origin, thereby better satisfying practical engineering requirements. Motivated by the above analysis, this paper investigates the practical consensus problem for two-layer leader-following multi-agent systems, where uncertain communication environments induce bounded time-varying deviations in coupling weights.

We study the minimum imbalance from three sides.

In this paper, we give a complete classification of all types of finite edge-primitive $2$-arc-transitive Cayley graphs and bi-Cayley graphs.

Our key proof ideas center around non-polarity of the distributions of local assignments, probed via a random exposure of elements.

On the methodological front, this paper proposes a “cutting domain” graph-theoretic segmentation framework.

This study develops R4-Graph, a four-phase sequence for sustainability-oriented discrete mathematics: Represent, Remove, Recover, and Redesign.