80,937 papers in this slice of arXiv.
Tuong Le, Son Nguyen
Knutson and Tao's hives is a combinatorial model to compute Littlewood--Richardson coefficients. Similar to hives, Speyer introduced skeps and used them to prove a Schur log-concavity conjecture by Lam--Postnikov--Pylyavskyy. We first introduce skew hive and skew skep models, which specialize to both hives and skeps, and use this to prove a skew Schur log-concavity result generalizing Lam--Postnikov--Pylyavskyy conjecture. As a consequence, we obtain some log-concavity results concerning Newell--Littlewood numbers and shadow skew Schur functions. Finally, we explain bijections between (skew) hives, (skew) skeps, and peelable tableaux by Nguyen--Nguyen--Woodruff, answering Speyer's question.
Feng Liu, Shuang Sun, Yan Wang +2
A graph G is perfectly weight divisible if, for every positive integral weight function on V(G) and every induced subgraph H of G with at least one edge, the vertex set V(H) can be partitioned into two sets A and B such that H[A] is perfect and the maximum weight of a clique in H[B] is smaller than the maximum weight of a clique in H. Perfect divisibility and its weighted form provide a natural approach to polynomial χ-boundedness. A fork, also known as a chair, is the graph obtained from a claw by subdividing one of its edges once. In this paper, we prove that every fork-free graph is perfectly weight divisible. As a consequence, we confirm a conjecture of Sivaraman that every fork-free graph is perfectly divisible.
Kevin Guan
In the transversal achievement game on the n×n board, two players alternately claim cells, and the first to own a transversal---a set of n cells of which no two share a row or column---wins. Ranđelović showed that the first player wins for every n≥4, while the game is a draw for n=2,3. We give an independent proof that the first player wins for n≥4 that additionally establishes a bound on the length of the win: the given strategy forces a win by ply 2n+3, i.e.\ on the first player's (n+2)-nd move, for every n≥4. The proof yields a strategy that is fully determined by a fixed rule on the current position and can thus be implemented directly. We isolate the use of the hypothesis n≥4 to two steps in the analysis, explaining why the argument fails at n=3. An exhaustive computational search implementing the strategy verifies it against every legal defense for n=4,5,6, confirming both the strategy's validity and that the 2n+3 bound is attained in these cases. The main theorem has also been formalized and machine-checked in Lean 4.
Milan Tenn
We prove the conjecture of Donnelly et al. that a certain class of Littlewood-Richardson tableaux are equinumerous with peakless Motzkin paths of length n. Furthermore, we construct an explicit bijection between this class of tableaux and peakless Motzkin paths of length n for all n≥1.
Radu Curticapean, Daniel Neuen, Amir Nikabadi +2
Two graphs G and H are homomorphism indistinguishable over a graph class F if they admit the same number of homomorphisms from every graph in F. A wide range of relaxations of graph isomorphism arise this way: isomorphism itself over the class of all graphs [Lovász, Acta Math. Hung. 1967], equivalence under the k-dimensional Weisfeiler-Leman algorithm over the graphs of treewidth ≤k [Dvořák, J. Graph Theory 2010], and quantum isomorphism over planar graphs [Mančinska-Roberson, FOCS 2020]. Since the class F is typically infinite, it is not clear a priori whether homomorphism indistinguishability over F is decidable; for planar graphs it is undecidable. Every class for which decidability was previously known is sparse. We give the first decidability results for dense graph classes: We introduce the dense Weisfeiler-Leman algorithm that decides homomorphism indistinguishability over the class of graphs of cliquewidth ≤k, the dense counterpart of treewidth. This relation was not previously known to be decidable. The algorithm colors k-tuples of vertex subsets rather than k-tuples of vertices. Beyond the class of all graphs of cliquewidth ≤k, we prove a general meta-theorem: homomorphism indistinguishability over every CMSO1-definable graph class of bounded cliquewidth is decidable, in randomized exponential time. For classes of bounded linear cliquewidth the bound improves to PSPACE, and we show this is tight by exhibiting such a class for which the problem is PSPACE-complete. These are the first general algorithms for homomorphism indistinguishability over dense graph classes.
Luc Lapointe, Luis Pena
The ring Rm of m-symmetric functions consists of the formal power series that are symmetric in the variables xm+1,xm+2,… but carry no symmetry in the first m variables. We develop a combinatorial theory for the specialization at t=0 of the Schur functions of Rm. Our main tool is a new characterization of right key tableaux as suprema of the sets of decreasing subwords of the reading words of the subtableaux of T. Being invariant under elementary Knuth transformations, this characterization is compatible with the RSK correspondence. We obtain in this way a generating function over semistandard tableaux for the m-symmetric Schur functions at t=0, together with a combinatorial proof of a Cauchy identity in Rm. The m-symmetric Schur functions and their dual are then respectively identified with Demazure atoms and Demazure characters. Restricted to the last m variables, our correspondence specializes to a proof, by ordinary RSK, of Lascoux's nonsymmetric Cauchy identity for Demazure characters and atoms. As further applications, we relate the m-symmetric Schur functions at t=0 to the almost symmetric Schur functions through a unitriangular change-of-basis matrix, obtain tableau generating functions and Cauchy identities for both families, and derive Jacobi-Trudi type determinantal formulas for three different bases.
Kai Liang
This study applies finite automata to the automatic solving of a variety of combinatorial games. For games whose positions and moves can be represented as regular languages and their operations, we design a two-stage automatic solving algorithm: first, construct a candidate finite automaton to determine the P- and N-positions, and then perform rigorous formal verification on this automaton; once verified, a complete solution of the game is obtained. For partizan octal games, we introduce a generalized misère quotient, overcoming the limitation that traditional theory applies only to impartial games. Using the above algorithm, we successfully solve the majority of two-digit partizan octal games, and based on these results, we propose a partizan version of Guy's conjecture. We also successfully solve a considerable number of partizan octal games under misère play, and give a conjecture on the structure of those games exhibiting ``algebraic periodicity'' among them. For Kotzig's nim, we resolve the most important related conjecture: we prove that the outcomes and SG values are periodic under both normal and misère play (including their partizan versions). Our algorithm successfully solves several small-scale cases, including misère play and partizan versions. This study pioneers a new theoretical tool and algorithmic paradigm for the automatic solving of combinatorial games, and has broad prospects for further extension and application in the field of combinatorial game theory.
Krishnarjun Krishnamoorthy
We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.
Peiru Kuang, Yan Wang
For an integer n≥0, let f(n) be the minimum number of subcubes of Z3n of the form A1×⋯×An, where ∣Ai∣=2 for every i, whose union covers Z3n. A simple counting argument gives f(n)≥(3/2)n, while f(n)=O(n(3/2)n) by random construction. We prove that f(n)≤2(3/2)n−1, answering a problem of Imre Leader. We also show that f(n)/(3/2)n is nondecreasing and there exists a constant C3 such that f(n)=(C3+o(1))(3/2)n where 1.62227<C3≤2.
Krishna Menon
Chapoton introduced an interesting family of polytopes called arbor polytopes, where a polytope Qτ is associated to any arbor τ. Chapoton conjectured that the polynomial h(τ) which counts lattice points in Qτ by number of nonzero entries is palindromic and unimodal. Athanasiadis, Xiao, and Yan recently proved that h(τ) is gamma-positive for a certain class of arbors they call octopuses. Since their proof was computational, they asked for one that is bijective. We present such a proof. We also extend their results by showing that h(τ) is gamma-positive for a larger class of arbors we call lopsided octopuses.
Ian Short, Margaret Stanier, Matty van Son +1
The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway--Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.
Zhi-Hong Sun
In this paper, we first extend the Christoffel-Darboux formula for orthogonal polynomials to general three-term recurrence sequences, and then investigate the identities and congruences for gn(x) and vn(x) given by align* &g₀(x)=1,\ g₁(x)=x+12,\ (n+1)²g_n+1(x)=(2n(n+1)+x+12)g_n(x)-n²g_n-1(x)\ (n≥ 1), \&v₀(x)=1,\ v₁(x)=x,\ (n+1)³v_n+1(x)=(2n+1)(n(n+1)+x)v_n(x)-n³v_n-1(x)\ (n≥ 1).align*
Chaya Keller, Shakhar Smorodinsky
A set in Rd is s-convex if it is the union of at most s convex sets. A family F satisfies the (p,q) property if among any p sets in F, some q intersect. Let HDd(s)(p,q) be the minimum number of points needed to pierce a finite family of s-convex sets that satisfies the (p,q)-property. Alon and Kalai (1995) proved that HDd(s)(p,q) exists for any p≥q≥d+1 and any s≥1, but the quantitative bounds they obtained are very loose. We present several improved upper and lower bounds, for a general d and for s-intervals of the line (i.e., HD1(s)(p,q)). In particular, we prove the following: (i) For every d≥2, s≥1 and δ>0, if p>q and q≥Cdlog(esp), then HDd(s)(p,q)≤p−q+1+Od,δ((s⋅qp⋅logqesp)ρd+δ), where ρd<d is the exponent in the weak epsilon-net theorem of Rubin (2022). (ii) For s≥1, p≥q≥2 and q≥C0slog(2s)log(ep), p−q+s≤HD1(s)(p,q)≤p−q+2s+1. This result provides the first near-tight estimate for HD1(s)(p,q) for q>2. (iii) For any fixed s, there are an integer κs∈{s,…,2s} and constants Cs,ps>0 such that, whenever p≥ps and q≥Cslog(ep), HD1(s)(p,q)∈{p−q+κs,p−q+κs+1}. Interestingly, this two-value concentration result holds, although the exact value of the threshold remains unknown. (iv) For any s≥1, HD3(s)(p,4)≥sp2−o(1). Already for families of convex sets, this significantly improves the best known lower bound on HDd(1)(p,d+1), for all d≥3.
Zhen Liu, Chuanshu Wu
For graphs G and H, let N(G,H) denote the number of unlabeled, not necessarily induced copies of H in G, and let NP(n,H) be the maximum of N(G,H) over all n-vertex planar graphs G. We prove that, for every fixed integer m≥3, NP(n,C2m+1)=2m(mn)m+Om(nm−1/5). Heath, Martin, and Wells reduced the determination of the leading term to a weighted optimization conjecture involving cycles and paths. We prove a stronger sharp cycle--path inequality for probability weights on the edges of a complete graph and characterize equality in their conjectured inequality. Together with their reduction lemma, this settles the conjecture and yields the formula above, including the stated error term. The cases m≥5 are new; combined with the known results for C3 and C5, this determines the leading term for every fixed odd cycle in planar graphs.
Nived J. M., Mathew C. Francis
We consider the problem of shifting two tokens placed on nonadjacent vertices u,v of a graph G on n vertices to two nonadjacent vertices u′,v′ of G using a sequence of token movements. In each step, a token is moved from the vertex it is on to a neighbour of that vertex, ensuring that the tokens remain on nonadjacent vertices after this move. We answer a question of Briański, Felsner, Hodor, and Micek [``Reconfiguring Independent Sets on Interval Graphs'', MFCS 2021] by showing that if the two tokens can be moved from their initial position to their final position, then it can be done using at most 4n moves.
Paul Obernolte, Jakub Dranczewski, Yixiu Yin +10
Physical neural networks perform learning through the intrinsic nonlinear dynamics of matter. Optimising their design presents a considerable challenge: complex many-body physics can provide powerful computation, but are expensive to simulate and large experimental optimisation runs are impractical to fabricate. Hence, the high-dimensional space of possible network topologies cannot be effectively directly searched. Here, we show that this search can be efficiently performed in an abstract graph space that is vastly cheaper to explore. Using random lasing networks -- composed of interconnected nanoscale waveguides and hosting strongly coupled lasing modes -- as an exemplar physical vision system, we establish a quantitative three-layer link: simple graph-theoretic metrics predict the nonlinear lasing physics, which in turn predicts vision performance. After validating this relationship using physical simulations, we exploit it to drive an evolutionary algorithm using graph metrics, producing network topologies that outperform random designs at a fraction of the computational cost (3000× speed-up compared to physical simulation). On simulated image-classification tasks, graph-optimised networks substantially improve classification accuracy. As our framework operates on network topology rather than substrate-specific physics, we anticipate it can transfer to other network-based physical learning systems, providing an efficient route for the directed design and optimisation of complex, strongly-interacting physical neural networks.
Mikhail P. Golubyatnikov
A graph Γ is called a Deza graph with parameters (n,k,b,a) if it has exactly n vertices, is k-regular, and for any two distinct vertices u and v, the number of common neighbors of u and v is either a or b. The graphs Δ1 and Δ2, which have the same vertex set as Γ, and in which two vertices are adjacent if they have a or b common neighbours, respectively, are called the children of the Deza graph. If, for a Deza graph Γ, both Δ1 and Δ2 are strongly regular graphs, then Γ is called a strongly Deza graph. In this work, we present a construction of an infinite family of strongly Deza graphs, for which the children Δ1 and Δ2 are strongly regular graphs with the same parameters as the graphs NOnε⊥(5) and NOnε⊥(5).
Igor Kleiner, David Perry
We study a cooperative search game with N lockers, N−1 labelled objects, one empty locker, and N−1 players. Player i seeks object i and may inspect at most two lockers; the second inspection may depend on the content of the first. The players may coordinate beforehand but receive no information about the searches of other players after play begins. We prove that the maximum probability that every player finds the assigned object is IN/N!, where IN is the number of involutions of N elements. An optimal strategy represents the blank by the common fictitious symbol N and follows pointers: player i first opens locker i, then opens the locker whose number was observed. The upper bound holds for every deterministic or randomized adaptive strategy and follows from a deletion lemma and two recurrences. We give explicit examples, identify the three-door case with an isomorphic "Return of Monty Hall" game, and report a reproducible mixed-integer verification for N=2,3,4,5. The computation is independent of, and not needed for, the proof.
Chengrui Fang, Jianfeng Hou
For an n-vertex graph G and a permutation π of its vertex set, let IG(π)=∣E(G)∩E(πG)∣, and let μ(G)=minπIG(π). Let f(n,k) be the minimum number of edges in an n-vertex graph G satisfying μ(G)≥k. Erdős recorded a construction of Mullin showing f(n,5)≤2n−2 and asked whether equality holds for sufficiently large n. We prove that it does: f(n,5)=2n−2 for all sufficiently large n. Equivalently, every sufficiently large n-vertex graph with at most 2n−3 edges admits a relabelling with at most four common edges. The proof combines a quantitative exclusion of almost-universal vertices, a finite high-degree core with low-degree buffer vertices, list packing, and a sparse permutation version of the Lovász local lemma.
Yair Lavi
The Foregger-Sinkhorn tie-point conjecture asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero is larger than its permanent, then that zero is a tie point. We give a counterexample in dimension eight.