Papers

This study is the $3^{rd}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10} and is a continuation of Part 2. Here, we obtain all the 384 pairs of Type-2 isomorphic circulant graphs of order 32.

This paper presents a generalization of the Kostant game, a combinatorial framework originally for generating positive roots in Lie algebras. By introducing an arbitrary multi-vertex modification, we prove that the resulting game configurations naturally biject with the minimal length representatives of parabolic quotients W/W_J. This yields a dynamical and algorithmic perspective on reduced words. Finally, we apply this framework to derive a novel root counting identity, formalize the Coxeter-theoretic foundation for combinatorial approaches to the Mukai conjecture, establish the regularity of reduced word languages via finite state automata, and dynamically construct Standard Young Tableaux.

The necklace polynomials \[ M_n(x)=\frac1n\sum_{d\mid n}\mu(d)x^{n/d} \] play a central role in discrete mathematics: they count aperiodic necklaces, enumerate monic irreducible polynomials over finite fields, and give the dimensions of homogeneous components of free Lie algebras. Despite their inherently discrete origins, we show that treating $M_n(x)$ as a function of a real variable $x$ unlocks surprising structural properties that answer natural enumerative questions. In this paper, we study $M_n(x)$ as a real-variable function and establish several new analytical and monotonicity properties. We prove that the normalized functions $M_n(x)/x^n$ and their higher normalized derivatives are strictly increasing on $[1,\infty)$. As a consequence, we show that the proportion of irreducible polynomials of fixed degree over $\mathbf F_q$ increases with $q$. We also establish strict growth with respect to the degree $n$ for $x\ge2$. In addition, we determine a sharp threshold for log-convexity: the sequence $\{M_n(x)\}_{n\ge2}$ is uniformly log-convex if and only if $x>8$. These results reveal unexpected analytic structure underlying necklace polynomials and show how real-variable methods can yield new information about discrete enumeration problems. For instance, it is shown that adding one more bead to a sufficiently long necklace will approximately increase the total number of primitive, rotationally distinct configurations by a factor of the number of available colors.

This study is the first part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. Circulant graphs $C_n(R)$ and $C_n(S)$ are said to be Adam's isomorphic if there exist some $a\in \mathbb{Z}_n^*$ such that $S = a R$ under arithmetic reflexive modulo $n$ \cite{ad67}. In this paper, the author modified his earlier definition \cite{v96} of Type-2 isomorphism w.r.t. $m$ such that $m$ and $m^3$ are divisors of $\gcd(n, r)$ and $n$, respectively, and $r\in R$. Using the modified definition, we present our study on Type-2 isomorphism of circulant graphs $C_n(R)$ w.r.t. $m$ = 2. We prove that $(i)$ $C_{16}(1,2,7)$ and $C_{16}(2,3,5)$ are Type-2 isomorphic w.r.t. $m$ = 2; $(ii)$ For $n \geq 2$, $k \geq 3$, $1 \leq 2s-1 \leq 2n-1$, $n \neq 2s-1$, $R$ = $\{2, 2s-1, 4n-(2s-1)\}$ and $S$ = $\{2, 2n-(2s-1), 2n+2s-1\}$, $C_{8n}(R)$ and $C_{8n}(S)$ are Type-2 isomorphic w.r.t. $m$ = 2, $n,s\in\mathbb{N}$; and $(iii)$ For $n \geq 2$, $1 \leq 2s-1 < 2s'-1 \leq [\frac{n}{2}]$, $0 \leq t \leq [\frac{n}{2}]$, $R$ = $\{2,2s-1, 2s'-1\}$ and $n,s,s'\in \mathbb{N}$, if $\theta_{n,2,t}(C_n(R))$ and $C_n(R)$ are isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 for some $t$, then $n \equiv 0~(mod ~ 8)$, $2s-1+2s'-1$ = $\frac{n}{2}$, $2s-1 \neq \frac{n}{8}$, $t$ = $\frac{n}{8}$ or $\frac{3n}{8}$, $1 \leq 2s-1 \leq \frac{n}{4}$ and $n \geq 16$ where $\theta_{n,m,t}$ is a transformation used to define Type-2 isomorphism of a circulant graph. At the end, we present a VB program POLY215.EXE which shows how Type-2 isomorphism w.r.t. $m$ = 2 of $C_{8n}(R)$ takes place for $R = \{2, 2s-1, 4n-(2s-1)\}$, $n \geq 2$ and $n,s\in {\mathbb N}$.

We study test sets: subfamilies of sequences converging to a point P that still suffice to detect every discontinuity of real-valued functions at P. Ordered by inclusion, these test sets form a poset. Under natural hypotheses at P, we prove that this poset has a minimal element. We also analyze its maximal chains, showing that some have a least element, while others do not. Finally, on the sequential fan we give a concrete realization in which the minimal test set produced by our construction has strictly smaller cardinality than the full family of convergent sequences.

For the OEIS sequence A001711, contributed by N. J. A. Sloane long before the on-line era and identified there as the diagonal $T(n+4, 4)$ of a generalized-Stirling triangle, R. J. Mathar contributed in February 2020 the conjectured order-2 P-recursive recurrence \[ a(n) - (2n+5)\,a(n-1) + (n+2)^{2}\,a(n-2) \;=\; 0,\qquad n \ge 2. \] We give a one-page proof. Detlefs's harmonic-number closed form $a(n) = \tfrac{1}{4}(n+3)!\,(2 H_{n+3} - 3)$ collapses the left-hand side, after dividing through by $(n+1)!/4$, to a polynomial identity of $n$ with coefficient $H_{n+2}$. The harmonic-number coefficient simplifies to $(n+3) - (2n+5) + (n+2) = 0$ (using $H_{n+3} = H_{n+2} + \tfrac{1}{n+3}$ and $H_{n+1} = H_{n+2} - \tfrac{1}{n+2}$); the constant remainder is $-3 \cdot 0 = 0$ for the same reason. The supplementary archive contains a SymPy script verifying both pieces symbolically, the e.g.f.\ expansion against the harmonic closed form, and Mathar's recurrence numerically for $n = 2, \ldots, 5000$.

Edge ideals of graphs were introduced by Villarreal in 1990, and have been the subject of many studies since then. In the same year, Fröberg characterized edge ideals with regularity 2 in combinatorial terms. This result was generalized by Fernández-Ramos and Gimenez to regularity 3 for bipartite graphs. A key ingredient in these results is the particular shape of the Betti diagrams of the edge ideals of the graphs obtained after removing a Hamiltonian cycle from either a complete graph $ K_k$ or a complete bipartite graph $K_{k,k}$. In this work, we consider the family of Generalized Andrásfai graphs ${\rm GA}(t,k)$ with $t\geq 1 $ and $k \geq 2$. This family extends the families of complete graphs, since $K_{k+1} = {\rm GA}(1,k)$, and complete bipartite $k$-regular graphs, since $K_{k,k} = {\rm GA}(2,k)$. We show that the results known for $ K_k$ and $ K_{k,k}$ can be naturally extended to this family. More precisely, when removing a suitable Hamiltonian cycle from ${\rm GA}(t,k)$, the resulting edge ideal has regularity $t+2$, projective dimension $t(k-2)$ and a Betti diagram exhibiting a generalized version of the same special shape.

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the first minimum-degree condition which is polynomially smaller than linear. Our methods yield an analogous result when the host graph is not required to contain a Hamilton cycle, but only a $2$-factor consisting of at most $k$ cycles; this answers a question of Bucić, Jahn, Pokrovskiy and Sudakov.

For an integer $m\ge 1$ and a parameter $\alpha>0$, consider the nested recursion $$ Q_{\alpha,m}(n)=1+\left\lfloor \frac{\alpha}{m}\sum_{j=1}^m Q_{\alpha,m}\!\left(n-Q_{\alpha,m}(n-j)\right)\right\rfloor, \qquad n>m, $$ with $Q_{\alpha,m}(1)=\cdots=Q_{\alpha,m}(m)=1$. For $m=1$ and $\alpha=1$, this is Golomb's non-homogeneous triangular recursion. We prove that its canonical triangular solution is preserved, up to an initial index shift, by every finite arithmetic averaging length. More generally, the same exact solution is generated by any aggregator satisfying a local floor-lock condition. This class includes all power means of finite order, including the harmonic and geometric means, as well as the minimum and positively weighted quasi-arithmetic means. For $m\ge 2$, the maximum lies outside this class but has a different explicit block law. Consequently, every value $k\ge 2$ occurs exactly $k$ times in the floor-admissible class, and $$ Q_{1,m}(n)=\left\lfloor\frac{1+\sqrt{1+8(n-m)}}{2}\right\rfloor \sim\sqrt{2n}. $$ For $0<\alpha<1$, the sequence is identically one. Near criticality, with $\alpha=1+\delta$ and $0<\delta<(2m-1)^{-1}$, we determine the exact first departure time from the critical orbit, of order $\delta^{-2}$. We also prove a finite-step breakdown criterion for large $\alpha$ and a conditional slope theorem: any globally defined solution with a limiting density in $(0,1)$ must have slope $1-\alpha^{-1}$. Exact-arithmetic computations support, but do not prove, a supercritical linear-growth regime.

We prove several types of Courant nodal domain theorems for generalized Laplacians on graphs, based on an invariant introduced by Urschel, which we call the "Urschel number", denoted ${\rm UN}({\bf f})$, of an eigenvector ${\bf f}$. We refine Urschel's invariant, and use perturbation techniques to obtain some new results. First, we show the existence of mutually orthogonal eigenvectors, such that if the $k$-th eigenvalue has multiplicity $m$, then for $0\le j\le m-1$, ${\rm UN}({\bf f}_{k+j})\le k+\min(j,(m-1)-j)$. Second, for a simple $k$-th eigenvalue, we classify the zeroes of ${\bf f}_k$ as either "shallow or "deep"; we obtain a number of results that say, roughly speaking, the more shallow vertices ${\bf f}_k$ has, the more control we have over our new invariants based on Urschel's. Our new invariants of an eigenvector, ${\bf f}_k$, are a sequence of integers whose minimum value is ${\rm UN}({\bf f}_k)$ and whose maximum, denoted ${\rm UN}_{\max{}}({\bf f}_k)$, is the maximum number of nodal domains of any possible positive/negative signing or "charge" of the zeroes of ${\bf f}_k$. An example of our second type of result is that if ${\bf f}_k$ has no deep vertices, then ${\rm UN}_{\max{}}({\bf f}_k)\le k$. We provide a number of examples to illustrate our main results, and how they differ from the situation in analysis. We also describe a minor improvement of the Gladwell-Zhu theorem for an orthonormal eigenbasis in the presence of eigenvalues of sufficient multiplicity.

Balanced and well-balanced subsets of the set of positive roots of compact Lie algebras arise naturally in problems related to Hermitian and spin geometry. In this paper we compute the maximal and minimal size of well-balanced subsets in all simple root systems.

The alternated composition of $N=2p$ differential operators $w_j(x)\,\partial_x^p$ of strict order $p$ on the line $\mathbb{R}\ni x$ is again a differential operator of strict order $p$; its coefficient is the constant $\mathrm{const}(p)$, depending only on the arity $N$, times the Wronskian determinant of the originally taken coefficients $w_1,\dots,w_N$. The case $p=1$ of the Lie bracket for two vector fields fixes $\mathrm{const}(1)=1$, and $\mathrm{const}(2)=2$ is found easily by hand; $\mathrm{const}(3)=90$ can still be obtained symbolically. The problem is to determine $\mathrm{const}(p\geqslant4)$. We compute $\mathrm{const}(p)$ exactly for all $p\leqslant14$ – a 241-digit integer at $p=14$ – and record the resulting integer sequence as OEIS A392714. We prove that $v_p(\mathrm{const}(p))\geqslant p-1$ for every prime $p$, matching the exact equality observed numerically throughout our range, and conjecture that this equality holds in general. We show that $\log\mathrm{const}(p)$ grows like $\alpha p^2\log p$, with the leading coefficient close to $2$, nearly saturating the bound $p^2\log p\,(1+O(1/\log p))\leqslant\log\mathrm{const}(p)\leqslant2p^2\log p\,(1+O(1/\log p))$ obtained by O. Zaboronski (private communication). A naturally arising reduced constant is found to decay to zero super-exponentially rather than grow, a direct consequence of this near-saturation.

Menger's Theorem is a fundamental result in graph theory. It states that if in a graph $G$ with distinguished sets of terminal vertices $S$ and $T$ there are no $k$ pairwise vertex-disjoint $S$-$T$ paths, then there is a set of less than $k$ vertices that intersects every $S$-$T$ path. In this work, we give a coarse variant of this result for planar and bounded genus graphs. Precisely, we prove that for every surface $\Sigma$ there is a function $f\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N}$ such that for every pair of integers $d,k\in \mathbb{N}$ and a $\Sigma$-embeddable graph $G$ with distinguished sets of terminal vertices $S$ and $T$, if $G$ does not contain a family of $k$ $S$-$T$ paths that are pairwise at distance larger than $d$, then there is a set $X$ consisting of at most $f(d,k)$ vertices of $G$ such that every $S$-$T$ path is at distance at most $d$ from a vertex of $X$. This partially answers questions of Nguyen, Scott, and Seymour [arXiv:2508.14332], who proved that such a result cannot hold in general graphs. A key ingredient of our proof is a structure theorem from the developing ”colorful” graph minor theory, where the focus is on studying the structure in a graph relative to some fixed subsets of annotated vertices. In our case, these annotated vertices are $S$ and $T$.

Folding subgroups give a way to realize non-simply-laced Coxeter groups as subgroups of simply-laced Coxeter groups. In this paper, we study how folding subgroups of finite and affine type are distributed length-wise by calculating the length generating function of the subgroup with respect the length of the ambient group. These generating functions have surprisingly nice formulas in terms of $q$-integers and give rise to interesting combinatorial identities on polynomials involving length statistics of both the ambient group and folding subgroup.

Let $M$ be a symmetric matrix over $\mathbb{F}_2$, and let $\operatorname{diag}(M)$ be its diagonal vector. It is known that $\operatorname{diag}(M)\in\operatorname{Im}(M)$. Thus the affine system $Mx=\operatorname{diag}(M)$ is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies $\operatorname{diag}(M)^T x\equiv\operatorname{rank}(M)\pmod{2}$. For graph matrices, this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices $A(G)+I$ to arbitrary partially looped graph matrices $A(G)+D$. We also study how rank and nullity change when loops are toggled. Algebraically, simultaneous loop toggling on the support of a vector $u$ is the diagonal rank-one update $M\mapsto M+uu^T$. We prove an exact three-case rank and nullity formula for this update. Finally, for rooted trees with arbitrary binary diagonal labels, we give a finite-state boundary recursion using affine subspaces of $\mathbb F_2^2$. This recursion counts all generalized odd-domination patterns and implies eventual quasigeometric nullity formulas for complete rooted trees with eventually periodic depth labels.

The Mayhew–Newman–Welsh–Whittle conjecture predicts that asymptotically almost all matroids are sparse paving. We study the gap between paving and sparse paving matroids at the logarithmic scale. Let \(p_n\) be the number of paving matroids on \([n]\), let \(sp_n\) be the number of sparse paving matroids on \([n]\), and let \(sp_{n,r}\) be the number of rank-\(r\) sparse paving matroids on \([n]\). We prove that \[ p_n-sp_n\ge sp_{n,\lfloor n/2\rfloor}^{1-o(1)}. \] Thus the paving matroids that are not sparse paving are themselves logarithmically large. The construction prescribes one hyperplane larger than the rank and then counts stable sets in an induced subgraph of a Johnson graph. We also give amplified versions obtained by varying the large hyperplane and by prescribing distance-six families of large hyperplanes.

Motivated by the Erdős–Sós bipartite link conjecture, Füredi (Oberwolfach, 2004) asked for the asymptotic maximum edge density $\pi_{\mathrm{link}}(t)$ of $3$-graphs in which the link graph of every vertex is $t$-partite. Goldwasser's recursive blow-up construction based on projective planes gives the lower bound $\pi_{\mathrm{link}}(t)\ge 1-t^{-1}-(2+o_t(1))t^{-2}$ whenever $t-1$ is a prime power. In this note, we prove the upper bound $\pi_{\mathrm{link}}(t)\le 1-t^{-1}-t^{-2}/12$ for every $t \ge 2$. Together with Goldwasser's construction, this determines, up to a constant factor, the correct order of the gap between $\pi_{\mathrm{link}}(t)$ and the trivial averaging upper bound $1-t^{-1}$ for all prime-power values of $t-1$. In fact, our argument applies in the more general setting of $3$-graphs with no generalized daisies, equivalently, $3$-graphs in which the link graph of every vertex is $K_{t+1}$-free. We also establish an analogous upper bound for the positive $(r-1)$-codegree Turán density of generalized daisies.

We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root–to–leaf path. We provide two complementary combinatorial frameworks. First, we introduce a word model: unlabeled spinal networks correspond to a suitable class of restricted words with fixed multiplicities, taken modulo a simple relabeling equivalence, which yields an explicit closed enumeration. Second, we develop a symbolic-method approach based on a marked version of trees that admits a clean recursive specification; its boxed-product translation leads to a solvable bivariate generating function and a direct derivation of the coefficients.

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap. We consider the Stochastic Block Model (SBM), where the graph has a planted partition with $k$ equal-size blocks which form the `communities', and where, for parameters $p>q$, vertices within the same community connect with probability $p$, while vertices in different communities connect with probability $q$, independently across pairs of vertices. Modularity–based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM. We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a `planted' model, as most such analyses to date consider the `null' model. As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is $o(n)$ local moves away from the planted partition, where $n$ is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.

We prove that any random vector in $\mathbb{R}^n$ which is dominated in convex order by a standard Gaussian vector can be written as the sum of three standard Gaussian vectors. This implies that any $1$-subgaussian random vector in $\mathbb{R}^n$ is the sum of a universal number of Gaussian vectors. It also solves M. Talagrand's convexity problem, which in turn implies a weak version of a combinatorial analogue to the problem.

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density above which every graph that admits a red-blue edge-coloring with no red copy of $H_1$ and no blue copy of $H_2$ has bounded chromatic number. We give a complete answer when both $H_1$ and $H_2$ are 3-chromatic. The threshold takes exactly one of the five values \[ \frac23,\quad \frac57,\quad \frac34,\quad \frac79,\quad \frac45, \] and we characterize precisely which pairs $(H_1,H_2)$ give each value. The classification is determined by the ordinary chromatic thresholds of $H_1$ and $H_2$ and by their embeddability into a hierarchy of $C_5$-type Ramsey configurations.

We study two closely related objects associated with plane domains bounded by rational algebraic arcs: canonical forms in the sense of positive geometry and normalized moment-generating functions, or Fantappie transforms. For polygons these objects are related by polarity: the normalized Fantappie transform of a polygon is the canonical form of the polar polygon. For genuinely curved polypols the same dual-geometric mechanism survives, but the transform is no longer a rational logarithmic canonical form; rather, it is a holonomic, generally branched period whose singularities are controlled by vertex hyperplanes and by the projective dual curves of the nonlinear boundary components. We give explicit examples, including sectors and half-disks, and explain how harmonic moment generating functions arise as one-dimensional restrictions of the same Fantappi`e transform.

Chromatic choosability is a notion of fundamental importance in list coloring. A graph $G$ is chromatic-choosable when its chromatic number, $\chi(G)$, is equal to its list chromatic number $\chi_{\ell}(G)$. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted $P_{\ell}(G,m)$, which is the list analogue of the chromatic polynomial of $G$, $P(G,m)$. A graph $G$ is said to be enumeratively chromatic-choosable when $P_{\ell}(G,m)=P(G,m)$ for every $m \in \mathbb{N}$. Theta graphs and their generalizations have played an important role in graph coloring problems over the years; for example, they appear in the characterization of chromatic-choosable graphs with chromatic number 2. In this paper we characterize the enumeratively chromatic-choosable theta graphs. Our proof utilizes ideas from DP-coloring (a.k.a. correspondence coloring), providing yet another example of how the more general setting of DP-coloring can be leveraged to attack a problem in list coloring.

Similarity metrics are central in the theory of large networks and graph limits. For bounded-degree graphs, the Benjamini–Schramm metric records the distribution of rooted neighbourhoods, while the stronger colored-neighbourhood metric gives rise to local-global convergence. In this paper we show that this intricate topology is already determined by much smaller observations. For technical convenience and greater generality, we work with graphings, which are measurable generalizations of finite graphs and include all finite graphs as special cases. We prove that, for graphings of uniformly bounded degree, convergence of all colored degree distributions, or equivalently of all colored star statistics, is equivalent to local-global convergence. We also introduce an even more economical sampling procedure, the colored cherry metric, in which one observes only the root and two randomly chosen neighbours, and prove that it induces the same topology. Thus the full local-global structure can be reconstructed, at the level of topology, from families of very small colored observations. Our star-observation theorem was previously announced in the work of Backhausz and the author as an important ingredient in the proof that the so-called action convergence unifies dense graph limit theory with local-global convergence, thereby providing a general graph limit theory for sparse, dense, and intermediate-density graphs.

The conflict-free closed neighborhood (CFCN$^*$) chromatic number of a graph $G = (V,E)$ is the smallest positive integer $k$ for which there exists a coloring of a subset of vertices using $k$ colors such that, for every vertex in $V$, there exists a color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) chromatic number is defined analogously. In this paper, we study `list variants' of the above-mentioned coloring parameters. The conflict-free closed neighborhood (CFCN$^*$) choice number of a graph $G = (V,E)$ is the smallest positive integer $k$ such that for every assignment of lists of size $k$ to its vertices, there exists a coloring of a subset of vertices, say $V'$, in which (i) every vertex in $V'$ receives a color from its list, and (ii) for every vertex in $V$ there exists some color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) choice number is defined analogously. Dębski and Przybyło [Journal of Graph Theory, 2022] showed that for any graph $G$ with maximum degree $\Delta$, the CFCN$^*$ chromatic number of its line graph is $O(\ln \Delta)$. This result was later extended to claw-free graphs by Bhyravarapu et al. [Journal of Graph Theory, 2025], who proved that every $K_{1,k}$-free graph $G$ admits a CFCN$^*$ coloring using $O(k\ln \Delta)$ colors. In this paper, we generalize this result to the list setting and show that every $K_{1,k}$-free graph $G$ has a CFCN$^*$ choice number of $O(k\ln \Delta)$. Further, we answer some questions concerning the hardness of computing CFCN$^*$/CFON$^*$ choice numbers posed by Gupta and Mathew [SOFSEM, 2026]; in particular, we show that it is NP-hard to determine whether the CFCN$^*$/CFON$^*$ choice number a graph is equal to $k$, for $k=1,2$.

Motivated by the change-of-domain problem for additive bases, Bukh, van Hintum and Keevash conjectured that if \(A,B\subseteq \mathbb{Q}^{n}\) and \(\{\boldsymbol{e}_i+\boldsymbol{e}_j:1\le i\le j\le n\}\subseteq A+B,\) then \(|A|+|B|\ge 2n\). They further proposed the strengthened conjecture: if \(|A|=n-t\), then \(|B|\ge n+\binom{t+1}{2}.\) Bukh also explicitly asked whether the same bounds hold for \(A,B\subseteq \mathbb{R}^{n}\) and an arbitrary basis \(S\) of \(\mathbb{R}^{n}\), under the assumption \(S+S\subseteq A+B\). We prove the full strengthened statement over \(\mathbb{R}^{n}\): if \(S+S\subseteq A+B\) and \(|A|\le n-t\) with \(0\le t\le n-1\), then \(|B|\ge n+\binom{t+1}{2},\) which is sharp for every basis \(S\) and every \(0\le t\le n-1.\) The proof is short, using edge contractions in a graph-theoretical framework and a new coloring lemma over \(\mathbb F_2^n\).

We argue that Jack Littlewood-Richardson coefficients $g_{\mu\nu}^{\lambda}(\alpha)$ are specialisations of certain novel polynomials. For the triple of partitions $(\mu,\nu,\lambda)=(21,21,321)$, we prove the corresponding polynomial is invariant under $S_6 \times \mathbb{Z}_2$, which is identified as the automorphism group of the Johnson graph $J(6,3)$. We conjecture that these polynomials exhibit a factorization property on certain hyperplanes, which is a consequence of compatibility relations between polynomials associated to adjacent triples in the Young graph. As a consequence of this, we conjecture that the difference of adjacent Jack Littlewood-Richardson coefficients is divisible by the shared hook length.

The regular set tolerance is an important term in sensitivity analysis. For combinatorial sum problems, e.g., the Traveling Salesman Problem, Shortest Path Problem and Minimum Spanning Tree Problem, it determines how much the sum of the costs of the elements of a set can be increased while ensuring that all current optimal solutions remain optimal. The regular set lower tolerance determines how much the sum of the costs of the elements of a set can be decreased while ensuring that the objective value of the optimal solution is not changed. We investigate general methods for computing regular (upper and lower) set tolerances in combinatorial sum problems. For the upper tolerance, we present a linear programming approach, and for the lower tolerances, three linear programming approaches, where the last two are novel and lead to recursive procedures for computation of the lower tolerances of all subsets of the given ground set. We give new upper bounds for set lower tolerances. For both upper and lower tolerances, we give an exact formula for sets of cardinality 2 and 3. Furthermore, we computationally compare and analyze the three different lower tolerance LPs. Finally, we consider the computation of tolerances for the Minimum Spanning Tree Problem, give a formula for single tolerances, a lower bound for regular set upper tolerances and an exact formula for regular set lower tolerances.

This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real variable $t$ in the sense that \[ L(P,t)=\sum_{k=0}^{n} c_k(P,t)t^k, \quad t\geq 0, \] where $c_k(P,t)$ are periodic piecewise polynomials of degree $n-k$ if ${\rm aff}\,P$ contains the origin, and are periodic functions vanishing almost everywhere otherwise. When $P$ is a rational simplex $\sigma$, the coefficient functions $c_k(\sigma,t)$ are given explicitly in terms of vertex information of the simplex $\sigma$. Moreover, the reciprocity law still holds.

We introduce the subsum polynomial of a partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ defined by $\mathrm{sp}(\lambda, x)=\prod_{i=1}^k(1+x^{\lambda_i})$. We study the sum of reciprocals of $\mathrm{sp}(\lambda, x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.