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In graph pebbling, pebbles are distributed across the vertices of a graph. A pebbling move consists of removing two pebbles from one vertex and adding one pebble to an adjacent vertex. The pebbling number is the minimum number $t$ such that, regardless of how the $t$ pebbles are initially placed, it is always possible to perform a sequence of moves that results in at least one pebble placed on any specified target vertex. Graphs whose pebbling number is equal to the number of vertices are called Class~$0$ and provide a challenging set of graphs that resist being characterized. In this note, we answer a question recently proposed by the pioneering study on the pebbling number of snark graphs: we prove that the smallest Flower snark $J_3$ is Class~$0$, establishing that $J_3$ in fact is the only Class~$0$ Flower snark. that systematically reduce the space of pebble configurations. Our bound-based approach shows particular promise for diameter $3$ graphs, where the neighborhood structure remains sufficiently constrained to enable effective analysis.

We prove that integer linear feasibility is strongly $NP$-complete even when the constraint matrix has exactly two nonzero entries in every row and column, with its support forming a single cycle. This holds already for monotone systems (one positive and one negative coefficient in each inequality) with nonzero coefficients of absolute value $2$ or $3$ and right-hand sides in $\{-1,0,1,2\}$. No variable bounds are supplied separately from the system. For systems in which each inequality involves at most two variables, this strengthens known hardness results by restricting all numerical data to fixed finite sets. For systems in which each variable occurs in at most two inequalities, this establishes, to the best of our knowledge, the first $NP$-hardness result for feasibility. We also give negative answers, unless $P=NP$, to questions on polynomial-time solvability of integer programs with bounded coefficients and bounded totally $Δ$-modular treewidth, and on fixed-parameter tractability of integer bound propagation with respect to coefficient and right-hand-side magnitudes.

A graph is rigid if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $d\ge 3$. This provides strong positive answers to a question of van der Zypen [https://mathoverflow.net/q/296483, https://mathoverflow.net/q/321108]. Further, we use our construction to show that every finite monoid is isomorphic to the endomorphism monoid of a regular graph. This solves a problem of Babai and Pultr [J. Comb. Theory, Ser. B, 1980].

Monadically dependent graph classes are those in which first-order logic cannot encode all graphs. They include all nowhere dense classes and all classes of bounded twin-width. We show that first-order model checking is fixed-parameter tractable on every monadically dependent graph class. Together with the known hardness result for hereditary classes that are not monadically dependent, this gives a complete characterization of the hereditary graph classes with fixed-parameter tractable first-order model checking. This resolves a central open problem in algorithmic model theory. Our proof combines ideas from structural graph theory (graph decompositions), finite model theory (locality of first-order logic), and model theory (counting types). 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 $U$.

We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $π_{n,λ}$ under which each partial triangulation $σ$ is assigned probability proportional to $λ^{|σ|}$, where $λ>0$ is a model parameter and $|σ| \in \{0,\dots,n-1\}$ denotes the number of diagonals in $σ$. This model belongs to a broad class of weighted geometric partition problems that include lattice triangulations and dyadic tilings, and is closely related to several classical combinatorial structures, including the full triangulations of a convex polygon and the associated Catalan structures. Our main result is a simple exact sampling algorithm for $π_{n,λ}$ with expected running time $O\big((\min\{n,n\sqrtλ\}+1)\log n\big)$, which is optimal up to the logarithmic factor.

The \textit{boxicity} of an undirected graph $Γ$, denoted as $\operatorname{box}(Γ)$, is the smallest non-negative integer $d$ such that $Γ$ can be represented as the intersection graph of axis-parallel rectangular boxes in $\mathbb{R}^d$. The \emph{power graph} of a group $G$, denoted as $\operatorname{Pow}(G)$, is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We determine the exact boxicity of power graphs of finite nilpotent groups. For a cyclic group of order $n=\prod_{i=1}^{r}p_i^{e_i}>1$, where the $p_i$ are distinct primes, we prove that its power graph has boxicity $\sum_{i=1}^{r}e_i-\max_i e_i$. For a finite nilpotent group $G$, let the number of noncyclic odd Sylow subgroups be denoted by $t(G)$. If $G$ has odd order or if $G$ has a cyclic Sylow $2$-subgroup, then $\operatorname{box}(\operatorname{Pow}(G))=\operatorname{box}(\operatorname{Pow}(\mathbb{Z}_{\operatorname{exp}(G)}))+t(G)$, where $\operatorname{exp}(G)$ denotes the \emph{exponent} of a group $G$. When the Sylow $2$-subgroup is noncyclic, and the odd part is nontrivial, the boxicity equals this value or one more, and we characterise precisely when each of these values is attained. The boxicity of power graphs of arbitrary finite groups and the exact boxicity of the comparability graph of the Boolean lattice were posed as open questions in Discrete Mathematics, 349(7):115017, 2026. As a consequence of our results, we answer the latter question completely and the former question partially, by computing the exact boxicity of the power graphs of nilpotent groups.

A natural way to compare two graphs is to align their vertex sets by a bijection and to measure the difference of the aligned adjacency matrices in a matrix norm. Depending on the norm, this yields, for example, the edit distance, the cut distance, the spectral distance, and the local edit distance. To make these distances applicable to graphs of different order, one can consider blowups of the graphs or, equivalently, relax bijections to fractional alignments, that is, optimal transport plans between the uniform distributions on the vertex sets, in the spirit of Gromov-Wasserstein distances. We investigate how much the optimal transport versions can differ from their bijective counterparts, that is, we quantify their integrality gaps. For the edit distance, we improve the best known upper bound on the integrality gap from 3 to 2 and the lower bound from 6/5 to 4/3. For the cut distance, we approach Lovász's conjecture that $δ_\square \le 2δ_\square^{\rm OT}$ from both sides. We improve the previously best known upper bound $δ_\square \le 32 \cdot (δ_\square^{\rm OT})^{1/67}$ (Borgs, Chayes, Lovász, Sós and Vesztergombi, Adv. Math. 2008) to $δ_\square \le C_\varepsilon \cdot (δ_\square^{\rm OT})^{1-\varepsilon}$ for every $\varepsilon > 0$, and we construct graphs $G,H$ with $δ_\square(G,H) = 2δ_\square^{\rm OT}(G,H)$, establishing a factor-2 lower bound that matches Lovász's conjecture. For the local edit and spectral distances, we obtain upper bounds of order $\sqrt{t\ln(1/t)}$ and $\sqrt{t}$, respectively, where $t$ is the corresponding optimal transport distance, and we show that, unlike for the edit and cut distances, no linear upper bounds are possible. Finally, we show that computing the optimal transport distances is NP-hard, and we give hardness of approximation results for the cut, spectral, and local edit distances.

A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. 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]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.

The Weisfeiler-Leman algorithms WL(k) for $k\ge1$ form a hierarchy of increasingly powerful tests for graph isomorphism. We study how the equivalence relations they induce on graphs transfer through a graph operation $Φ$: for a given $k$, what level of WL-equivalence between graphs $H_1$ and $H_2$ guarantees WL(k)-equivalence of graphs $Φ(H_1)$ and $Φ(H_2)$? We call the least such level the transfer parameter $τ_Φ(k)$ and determine it exactly for the line graph operation $Φ= L$: $τ_L(k)=2k+1$ for $k\ge2$, while $τ_L(1)=1$. We also extend these results to the $(r,S)$-line graphs, which generalize the line graph from edges to $r$-cliques by connecting two cliques when their intersection size lies in a prescribed set $S$. For these operations we prove that $τ_{r,S}(1)=r-1$ for every $S$ and, for certain choices of $S$, that $τ_{r,S}(k)=rk+r-1$ for $k\ge r$, and we bound the transfer parameter in the remaining range. Our upper bounds on transfer parameters follow from a treewidth analysis of diagrammatic expansions of homomorphism counts into $Φ(H)$ as linear combinations of homomorphism counts into $H$. The matching lower bounds combine explicit constructions of witness graphs with the homomorphism-distinguishing closedness of graphs of bounded treewidth.

We study the classical multi-item monopoly setting with a single additive buyer and $m$ heterogeneous items whose values are independent but not necessarily identically distributed. We compare the revenue (REV) of the optimal truthful auction, which in general may be randomized or complicated, with the revenues of three simple deterministic auctions: selling all items separately (SREV), selling them as a single grand bundle (BREV), and choosing the better of the two. Our technical cornerstone is a general upper bound on the optimal multidimensional revenue, parameterized by freely chosen item-wise caps, and expressed in terms of optimal single-item, Myersonian revenues and basic statistics of the value distributions. We derive this inequality through Lagrangian duality for a nonlinear programming formulation of the worst-case approximation ratio of separate-selling. Combining this inequality with carefully chosen caps and a probabilistic analysis of the resulting terms, we obtain improved approximation guarantees for all three deterministic auctions, both for general independent values and for iid values. Notably, we prove that $REV\le 2.5\max\{SREV,BREV\}$, improving the long-standing $5.2$ factor of Ma and Simchi-Levi and the $6$ factor of Babaioff, Immorlica, Lucier, and Weinberg. We complement this upper bound with a construction establishing a lower bound of $2.19$, improving the bound of $2$ by Rubinstein. For two iid items, we determine the exact approximation ratio of selling separately. Our proof constructs tight Lagrangian dual certificates for the aforementioned programming formulation, when item values lie in the grid $\{0,1/K,2/K,\dots,1\}$, for any discretization parameter $K$. Taking $K\to\infty$ yields the tight ratio $1+W(1/e)\approx1.2785$ in the continuous-valued setting, where $W$ is the Lambert-W function, closing the $[1.278,1.368]$ gap left by Hart and Nisan.