Papers

Slime Trail is a two-player combinatorial game in which the players alternately move a shared token to an adjacent vertex, permanently removing each vertex the token leaves, while attempting to reach a goal node. Ferland and Burke (2017) proved that Slime Trail is PSPACE-complete on arbitrary planar graphs and asked whether the same holds for the grid version actually used in play. We resolve this open problem by proving that Cardinal Grid Slime Trail, that is, Slime Trail on a square grid with four-directional movement, is PSPACE-complete. We adapt their QBF reduction to the grid setting, designing grid-compatible gadgets that respect the degree-4 bound and the parity constraints of the integer lattice. We further show the construction extends, under a 45-degree rotation, to the eight-directional variant.

The classical Durfee-square argument gives a combinatorial proof of the pentagon identity for the quantum dilogarithm. Just as the pentagon identity is associated with the $A_2$ quiver, Keller's identity is associated with the square-product quiver $A_2\square A_2$. Previous proofs of Keller's identity use cluster categories or spectral sequences in rapid-decay equivariant cohomology. We give three proofs of Keller's identity: a generating-function proof, an explicit weight-preserving bijection on colored partitions, and a standard-monomial proof using a four-colored arc algebra. Their common mechanism is an iterated Durfee decomposition: two possible pairings give horizontal and vertical decompositions, while a third binary step accounts for the coupling factor. These constructions provide a “superpotential analogue” of the Durfee-square argument.

In this paper, we study proper and complete edge-colorings of Johnson graphs $J(n,2)$, also called $n$-triangular graphs. They are isomorphic both to the 2-token graphs of complete graphs and to the line graphs of complete graphs. A $t$-edge-coloring of a graph $G$ is a function that assigns one color from $\{1,2,\ldots,t\}$ to each edge. Such a coloring is called proper if no two incident edges receive the same color, and complete if every pair of distinct colors appears on a pair of incident edges. The achromatic index, denoted by $\alpha_2(G)$, is the largest integer $t$ for which $G$ admits a proper and complete $t$-edge-coloring. We establish new lower and upper bounds for $\alpha_2(J(n,2))$, provide explicit proper and complete edge-colorings attaining the lower bounds, and determine the exact value of $\alpha_2(J(n,2))$ for several values of $n$.

A graph $\Gamma$ is $(c,t)$-sparse for $c > 0$ and $t \ge 1$ if for every pair of vertex subsets $A, B \subseteq V(\Gamma)$ with $|A|, |B| \ge t$, the number of edges $e(A,B)$ between them satisfies $ e(A,B) \le (1 - c)|A||B|$. In this paper, we prove that for every integer $\ell\ge2$, there are $\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $\Gamma$ is $(1-\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\ell}n^{1+1/\ell}$ edges, then $\Gamma$ contains at least $C'n^2t^{2\ell-2}$ induced copies of $C_{2\ell}$. This partially resolves a problem of Ding, Gao, Liu, Luan, and Sun.

In this paper, we study the dimension of the correlated equilibrium polytope of finite games. Under the oriented-matroid notion of genericity, we prove that if a generic game is not full-dimensional, then there exists a subgame whose correlated equilibrium polytope is affinely isomorphic to that of the original game. This settles and generalizes an earlier conjecture of Brandenburg, Hollering, and Portakal (2024). Moreover, we show that the existence of a correlated equilibrium whose slices are all non-zero implies that the correlated equilibrium polytope is either full-dimensional or a singleton.

Let $H$ be a fixed graph with $e(H)\ge3$ that contains two adjacent edges, and let $\ell\ge e(H)$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Writing $R_{H,\ell}(G)$ for the number of such colourings and $\lambda(H,\ell)$ for the rainbow entropy–Turán density on complete graphs, we show that, with high probability, the per-edge logarithmic counting rate can be made arbitrarily close to $\log\ell$ below a sufficiently small constant multiple of this scale, and arbitrarily close to $\lambda(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy–Turán parameter on complete graphs. We further investigate this parameter, obtaining partial exact evaluations, corresponding counting-stability results, and its first-order asymptotic behaviour as the number of colours tends to infinity. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph containing at least three edges, and provides a general mechanism for transferring complete-graph template entropy to sparse random hosts.

We prove that the set $\Pi_{\therefore,\infty}$ of uniform Turán densities of possibly infinite families of $3$-graphs contains a terminal interval: there exists $\delta>0$ such that $[1-\delta,1]\subseteq\Pi_{\therefore,\infty}$. Consequently, $\Pi_{\therefore,\infty}$ has positive Lebesgue measure and Hausdorff dimension $1$.

Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $\nu_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson–Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo–Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.

Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2023) conjectured that graphs of degree-$d$ polynomial growth can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size complete graph. Very recently, the case $d = 4$ of the conjecture was disproved by Illingworth, Norin and Steiner (2026). In this paper, we provide counterexamples to the conjecture for every integer $d \geq 2$, thus leaving $d=1$ as the only open case. Our counterexamples are appropriately subdivided cubic expanders. Our main contribution is to construct, for every real number $d>1$, subdivisions of cubic expanders with degree-$d$ polynomial growth and whose balanced separators have size $\Omega(n^{1-1/d}\log n)$, where $n$ denotes the number of vertices.

Given integers $r>t\ge1$ and a real number $p>0$, the $(t,p)$-norm $||\mathcal{H}||_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th powers of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T\subseteq V(\mathcal{H})$. When $t=r-1$, this is the codegree $p$-norm. For all sufficiently large $n$, we obtain the following results. The first two apply in both the convex range $p>1$ and the concave range $0<p<1$. First, for $r$-graphs with matching number at most $s$, we determine the maximum $(t,p)$-norm. Second, for $k$-intersecting families, we establish an Erdős–Ko–Rado-type theorem for the $(t,p)$-norm. Third, for $P_\ell^r$-free hypergraphs, we determine the maximum $(t,p)$-norm for every $1\le t\le r-1$ and $p>1$. In each of the three settings, we also characterize all extremal families.

A de Bruijn sequence is the cyclic prototype of a Cayley-graph observation problem: when does the ordered label word on a translated window $gY$ determine the vertex $g$? We distinguish three parameters. The unrestricted number $\operatorname{sep}_q(G)$ minimizes an arbitrary separating pattern; the connected number $\operatorname{csep}_q(G,S)$ requires a connected Cayley window containing $Y_S=\{1\}\cup S$; and the one-step number $\chi_1(G,S)$ fixes $Y_S$ and minimizes the alphabet. Thus $\operatorname{sep}_q$ is a group-level baseline, $\operatorname{csep}_q$ measures the cost of locality, and $\chi_1$ tests the smallest prescribed local window. The organizing theme is the tension between information and locality. Carbon tori test the gap between $\operatorname{sep}_q$ and $\operatorname{csep}_q$: for generalized dihedral groups $\mathbb{F}_{\ell^d}^{\times}\rtimes C_2$ we prove, for odd prime powers $\ell$, the sharp baseline $\operatorname{sep}_\ell=d+1$ and construct connected zig-zag windows, while the order-$14$ Heawood torus satisfies $\operatorname{sep}_4=2$ and $\operatorname{csep}_4=4$. The spherical $A_5$ example and a finite simple-group comparison test the fixed one-step window: explicit symmetric cubic generating tuples give $\chi_1(A_5,S)=3$ and $\chi_1(\operatorname{PSL}_2(\mathbb{F}_7),S)=4$, both at the counting bound, with structured matrix-coefficient certificates. Cyclic-coset packings, finite-field coordinates, and restricted matrix coefficients are used only as the construction tools these two examples require.

In this paper, we study extremal and stability problems for Berge Hamiltonian cycles in $r$-uniform hypergraphs under a minimum degree condition. Let $ g_r(n,t)=\binom{n-t}{r}+t\binom{t}{r-1}$, and let $t=t(k)$ be the unique integer satisfying $\binom{t-1}{r-1}<k\le \binom{t}{r-1}$. Using a sharp Pósa-type degree sequence theorem of Salia, we prove an extremal upper bound on the number of hyperedges in an $n$-vertex $r$-uniform hypergraph with minimum degree at least $k$ and with no Berge Hamiltonian cycle. We also prove a stability theorem in the dense range before the first minimizer of $g_r(n,t)$: every near-extremal example is contained in one of two natural non-Hamiltonian constructions.

We introduce a competitive two-player zero forcing game on a connected graph. Alice and Bob alternately seed white vertices or perform legal zero forces in their own colours, and each player seeks to minimise their own number of seeds. A force preservation rule prevents avoidable blocking of an opponent's established force. Because distinct continuations can be equally good for the player to move, optimal play is defined by a set-valued backward induction, and \(Z_g(G)\) is the minimum total number of seeds among the resulting optimal outcomes. We prove that \(Z_g(G)\geq Z(G)\), determine \(Z_g\) for paths, cycles, stars, complete graphs, and complete bipartite graphs, and characterise the graphs with \(Z_g(G)=2\) by an alternating two-chain forcing schedule. We also show that \(Z_g\) is not minor-monotone and that edge subdivision can either increase or decrease the parameter. Exact computation verifies \(Z_g(G)\leq2Z(G)\) through order nine.

With regard to the Hadwiger-Nelson problem, several 5-chromatic unit distance graphs in the Euclidean plane have been discovered in recent years. While most constructions rely heavily on the Moser spindle, a few recent examples completely avoid it, the smallest one consisting of 1441 vertices. In this note, we introduce an original geometric approach to constructing such graphs by utilizing the arcs of a 7-fold symmetric unit distance graph on 21 vertices, and obtain a Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices. While this is not a record small result, it arises from a straightforward, structured rule rather than a purely automated or brute-force search.

We present three proofs of the generalized Griesmer bound together with the corresponding proofs of the generalized antiGriesmer bound. The first proof follows from inequalities relating consecutive minimum and maximum subcode support weights. We also write the projective construction of Tsfasman and Vlăduţ as a residual code argument and express the geometric proof of Kurz, Landjev, and Rousseva in terms of shortened subcodes. In addition, complements in repeated simplex codes show that the two bounds are equivalent. The residual and shortening arguments also determine the consequences of equality for the resulting residual codes and shortened subcodes. Finally, the complement relation transfers known divisibility results for Griesmer codes to antiGriesmer codes.

We study edge-spectral supersaturation for two families of color-critical graphs with chromatic number three. For an integer $r\geq 1$, we define the spectral threshold \[ g_r(m):=\frac{r-1+\sqrt{4m-r^2+1}}{2}, \] which is the tight upper bound on the spectral radius of graphs avoiding $K_{s,t}^+$ (when $t+1\geq s\geq 3$) and $C_{2k+1}$ (when $r=k$), realized by split-graph constructions. First, let $t+1 \geq s\geq 3$ be fixed integers, and let $K_{s,t}^{+}$ be obtained by adding an edge to the part of size $s$ in $K_{s,t}$. We prove that every sufficiently large $m$-edge graph $G$ with $\rho(G)>g_{s-1}(m)$ contains $\Omega(m^{(s+t-1)/2})$ copies of $K_{s,t}^{+}$. Second, for any fixed $k\geq 2$, the condition $\rho(G)>g_k(m)$ forces $N(C_{2k+1},G)=\Omega(m^k).$ We also construct graphs showing that both lower bounds are tight up to constant factors. These results establish that exceeding the tight spectral Turán threshold $g_r(m)$ forces not just a single copy, but the optimal polynomial number of copies of these color-critical graphs. Thus, crossing the relevant split-graph spectral threshold forces the optimal polynomial order of copies, extending edge-spectral existence theorems to supersaturation results in the delicate three-chromatic regime.

Starting from an integer composition, form its consecutive absolute differences, provided that they are nonzero and pairwise distinct, and then permute these differences arbitrarily before repeating the operation. The depth of the composition is the maximum possible number of successive iterations. We determine the least positive integer admitting a composition of any prescribed depth. If a(n) is the least positive integer having a composition of depth n, then $a(n)=n+1+\lceil n(n+1)/4\rceil+\lfloor n/2\rfloor$. We also prove that every integer k at least a(n) has a composition of depth at least n. Consequently, if d(k) denotes the maximum depth of a composition of k, then $d(k)=\max\{n\geq 0:a(n)\leq k\}$. Finally, we classify and enumerate all compositions attaining the minimum a(n). Their number is given by four factorial formulas according to n modulo 4.

We study the simplicial boundary $\partial\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\partial\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial\mathcal{FS}\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\theta$-graphs. The result gives further evidence that $\partial\mathcal{FS}$ may be $(2n-3)$-spherical, an $\operatorname{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.

Let $W$ be an arbitrary Coxeter group. The shifted Bruhat interval $[w_1,w_2]\,x^{-1}$, the translate of the Bruhat interval $[w_1,w_2]$ by an element $x$, is partially ordered by the Bruhat order of $W$. These posets arise from affine pavings of Richardson varieties, and in general they are neither twisted intervals nor tilted Bruhat intervals. Our main result is that the shifted lower intervals $[e,w]\,x^{-1}$ are EL-shellable for every Coxeter group, via an explicit labeling of each cover by a reflection. Along the way we show that $[e,w]\,x^{-1}$ is a graded poset with a unique maximum given by the Demazure product and a unique minimum given by an opposite Demazure operator that we introduce.

A classical problem in convex and discrete geometry asks for the convex polyhedron of greatest volume whose vertices are chosen from the unit sphere $\mathbb{S}^2$. For a prescribed number $N$ of vertices, the problem is known only in a small number of cases. In this paper we resolve the next outstanding case, $N=9$. We prove that every convex polyhedron with at most nine vertices on $\mathbb{S}^2$ has volume at most $3\sqrt{2\sqrt{3}-3}$, with equality, up to rotation, precisely for a triaugmented triangular prism of an explicitly determined shape. The proof combines combinatorial and geometric reductions with sharp volume estimates. By a theorem of Berman and Hanes (Mathematische Annalen, 1970), a volume maximizer must be simplicial, reducing the $2,606$ combinatorial types of $9$-vertex polyhedra to $50$. We prove that a maximizer cannot have a trivalent vertex, leaving only five combinatorial types, which are treated using geometric and combinatorial arguments. In particular, we determine the exact maximizer within the triaugmented triangular prism class, and characterize the equality case.

By combining the recent results of Pierce and Xu on the higher-dimensional Burgess bound with the classical one-dimensional Burgess bound, we prove that for any sufficiently large prime $p$, there exists a Pythagorean triple $(a_p,b_p,c_p)$ with $a_p,b_p,c_p\in\mathbb{Z}\cap(0,p)$ such that $a_pb_p/2$ is a primitive root modulo $p$. This confirms a conjecture of Z.-W. Sun for all sufficiently large primes.

In light of Lovász's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, Du and Yuan considered a natural variant: what if vertex-transitivity is relaxed, while a high degree of symmetry–specifically edge-transitivity–is retained? To investigate this, they studied semisymmetric graphs (i.e. regular, edge-transitive, but not vertex-transitive graphs) and showed that every connected semisymmetric graph of order $2pq$, where $p$ and $q$ are distinct primes, contains a Hamilton cycle. In this paper, it is shown that for any prime $p$, every connected semisymmetric graph of order $2p^3$ also contains a Hamilton cycle.

For a graph $G$ of order $n$, let $\mathcal E(G)$ denote its adjacency energy and let $\alpha(G)$ denote its independence number. A recent theorem of Kumar and Pragada states that $$\mathcal E(G)\ge 2\bigl(n-\alpha(G)\bigr).$$ We determine all graphs attaining equality. More precisely, equality holds if and only if every connected component of $G$ is an isolated vertex, a balanced complete multipartite graph, or a graph obtained by taking the disjoint union of $K_{a,\ldots,a}$ and $K_{b,\ldots,b}$, with the same number $r\ge3$ of parts, and then completely joining corresponding parts.

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=\mu_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a transfer principle: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the total diagonal: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $\Delta_i$, the raw derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $\Delta_i$ extends holomorphically there, with $\Delta_i\phi=\partial_i^r\phi/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

We completely classify a natural scalar trace-product switching of the Gold almost perfect nonlinear function $x\mapsto x^3$ in every even dimension. Nontrivial switchings occur only for $n=4,6,8$: the admissible coefficients are, respectively, the nonzero trace-zero elements, the six elements of multiplicative order nine, and $\mathbb{F}_4^{*}$. For every even $n\geq10$, no nonzero coefficient is admissible. The infinite range is excluded by additive-character estimates on a Fermat cubic, with exact finite bridges for $n=10,12$. The raw coefficient lists for $n=6,8$ appeared earlier in Arshad's dissertation; our contribution is their intrinsic description, a proof uniform in the dimension, and the resulting dimension-rigidity theorem. We also classify normalized rank-two extensions in dimension eight by $\mathbb{P}^{1}(\mathbb{F}_4)$. A binary trace selector accepts two coefficient values at each non-base projective point, and the eight accepted marked switchings form exactly two extended-affine, hence two CCZ, classes. A centre-independent low-rank derivative criterion reduces each rank-$r$ candidate to $2^r-1$ membership tests in precomputed forbidden sets. The global APN classes reached are known; the results describe their local organization around the Gold centre and rule out this switching mechanism in all larger even dimensions.

The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, each vertex has a capacity, and the goal is to find a maximum weight subset of edges whose total incident demand at every vertex does not exceed its capacity. We study $(\alpha, \beta)$-bicriteria approximation algorithms, which return a solution of weight at least $1/\alpha$ times the optimum while allowing an additive capacity violation of at most $\beta$ times the maximum edge demand. We give an iterative relaxation algorithm for the demand matching problem that exploits a structural characterization of strictly fractional extreme points of the natural LP relaxation, which reduces the residual rounding problem to odd-cycle instances. Combined with a better-of-two rounding strategy, this yields $(7/6, 1)$- and $(1, 1)$-bicriteria approximation algorithms for general and bipartite graphs, respectively. We further generalize this approach to obtain a parametric family of algorithms, including a $(1, 4/3)$-bicriteria approximation. Separately, for the more general $k$-hypergraph demand matching problem, we give a greedy, combinatorial $(k, 1)$-bicriteria approximation algorithm. We complement these algorithmic results with matching lower bounds relative to the natural LP relaxation for $\beta = 0$ and all $\beta \geq 1$, completely characterizing the trade-off between weight approximation and additive capacity violation in this range.

For a natural number $n$ let $[n] = \{1,\ldots,n\}$. We say that a family ${\cal{S}}\subseteq 2^{[n]}$ is representing if every singleton set of $[n]$ is an intersection of some sets from ${\cal{S}}$. We show that the smallest possible cardinality of a representing set for $[n]$ is the discrete inverse $s(n)$ of the Sperner's function $n\mapsto \binom{n}{\lfloor n/2\rfloor}$, which by Sperner's Theorem is the maximum number of elements in an antichain in $2^{[n]}$ when viewed as subset (or boolean) lattice. Specifically, $s(n)$ is then the smallest positive integer such that $2^{[n]}$ contains an $n$-element antichain. Some generalization, further applications and asymptotics in terms of the second real branch of the Lambert $W$ function are presented.

We study subhomogeneity of the minimal operator system over a polyhedral cone, or equivalently the size of irreducible Arveson boundary points of free polyhedra. We obtain a complete classification in dimension three for cones (equivalently, dimension two for polytopes): cones with three extreme rays are 1-subhomogeneous, cones with four extreme rays are 2-subhomogeneous, and cones with at least five extreme rays are not subhomogeneous. In the last case, we construct irreducible Arveson boundary points at every even matrix level. We prove that subhomogeneity passes to faces and face quotients. We also determine the size of irreducible Arveson boundary points of free polyhedra over products of two simplices, obtaining a dichotomy between the product of two segments and all remaining cases. As applications, we classify n-dimensional cones with n+1 extreme rays. More generally, when every facet omits at most two extreme rays, we show that subhomogeneity occurs exactly for direct sums of simplicial cones and three-dimensional four-ray cones. We also construct, for a range of fixed dimensions and ray counts, cones exhibiting both subhomogeneous and non-subhomogeneous behavior.

In 2010, Levine found that the activity functions of parallel chip-firing games on the complete graph $K_n$ converge to a devil's staircase pattern: adding chips causes activity to progress through open intervals in which it is locally constant. In 2022, Bu, Choi, and Xu improved on an earlier bound by Kominers and Kominers to show that there exists a strict lower bound below which all games have activity $0$, and a strict upper bound above which all games have activity $1$. They thereby generalized the bottom and topmost rungs of the devil's staircase to all graphs. In 2024, Ji, Li, and Wang conjectured that a similarly general bound exists for games with activity $\tfrac12$. We use GPT-5.6-Sol to prove this conjecture, unifying existing results for trees, cycles, complete graphs, and complete bipartite graphs. This generalizes the middle rung of the devil's staircase.