Papers

Applying machine learning techniques to solving long-standing mathematical conjectures can be particularly challenging due to their extreme reward sparsity. As an illustrative example, we consider Kalai's algebraic Hirsch conjecture and recast the construction of its counterexamples as a sparse-reward reinforcement learning problem on graphs. We propose a constrained options-based HRL framework with an equivariant graph neural network policy, which allows us to learn useful temporal abstractions for this task. We evaluate our approach over a wide range of degrees and demonstrate that it consistently outperforms classical RL algorithms as well as greedy search. By exploiting the hierarchical structure of the problem, we effectively provide a first-of-its-kind application of HRL to a problem in commutative algebra.

The co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is defined to be a graph with the set of all non-trivial proper subgroups of $G$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. The deleted co-maximal subgroup graph of $G$, denoted by $\Gamma^*(G)$, is defined as the graph obtained by removing the isolated vertices from $\Gamma(G)$. In this paper, we prove that for any finite group $G$, $\Gamma^*(G)$ is connected. Furthermore, we show that $\Gamma^*(G)$ either contains a cycle or is a star. When $\Gamma^*(G)$ contains a cycle, its girth is either $3$ or $4$. Finally, we classify all finite groups $G$ for which $\Gamma^*(G)$ is a star.

Gross, Mansour and Tucker introduced the partial-twuality polynomials for ribbon graphs and investigated the interpolation property of these polynomials. The ribbon group generated by $\delta$ and $\tau$ acts on set systems as twist $\ast$ and loop complementation $\times$, yielding five nontrivial twuality operators: $ \{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast \}.$ Yan and Jin extended partial-twuality polynomials to set systems, yielding partial-$\bullet$ polynomials with $\bullet\in\{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast\}.$ For partial-$\ast$ polynomials, Zhao and Yan proved that this polynomial is either even, odd, or both even-interpolating and odd-interpolating for every binary delta-matroid. In this paper, we extend this interpolation property to all the remaining nontrivial partial-twualities of binary delta-matroids. Consequently, for every binary delta-matroid and every $\bullet\in\{\ast,\times,\ast \times ,\times\ast ,\ast \times \ast \}$, the partial-$\bullet$ polynomial is either even, odd, or both even-interpolating and odd-interpolating. We also provide examples to show that the binary assumption is essential.

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Turán number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash–Sudakov on the Turán density and the Turán number of the $4$-uniform expanded triangle. We also show that the Turán density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Turán density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.

In this paper, we study the signless Laplacian spectral radius of pure simplicial complexes under local homological restrictions on links. Let $K$ be a pure $r$-dimensional complex on $n$ vertices, ${\mathfrak q}_{r-1}(K)$ be the spectral radius of the $(r-1)$-up signless Laplacian of $K$, and ${\operatorname{lk}}_K(\sigma)$ be the link of a face $\sigma$ in $K$. We prove that if the homology $\widetilde H_t({\operatorname{lk}}_K(\sigma), {\mathbb R})=0$ for every face $\sigma\in K$ with $|\sigma|=r-t$, then \[ {\mathfrak q}_{r-1}(K)\le tn-(t-1)(r+1).\] Moreover, if $K$ is $r$-down path connected and $n\ge r+2+\binom{r+1}{t}\binom{r}{t}$, equality holds if and only if $K \cong \Delta_{r+1-t} \star \Delta_{n-r-1+t}^{t}$, where $\Delta_n$ denotes a simplex on $n$ vertices, $\Delta_n^{p}$ denotes the $(p-1)$-skeleton of $\Delta_n$, and $\star$ denotes the join of two complexes.

It is well-known that deleting or contracting any element of a connected matroid always yields at least one connected minor. However, for a connected polymatroid, only two such elements can be guaranteed, proved by Hall in 2013. This note investigates the connectivity properties of slice-projections of connected polymatroids, which includes deletion and contraction. We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the $j$-th slice-projection from the top of a polymatroid and $j$-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.

We study the inversion enumerator of permutations whose descent set is fixed to be the set of multiples of a fixed integer $k \ge 2$. For each $n$, let $\mathfrak{S}_n^{(k)}$ denote the set of permutations of $\{1,\dots,n\}$ whose descent set is exactly $\{i : k \mid i\}$, and define the polynomial $$ a_n^{(k)}(q)=\sum_{\sigma \in \mathfrak{S}_n^{(k)}} q^{\mathrm{inv}(\sigma)}. $$ We prove that the associated $q$-exponential generating function $$ F_k(t;q)=\sum_{n\ge 0}\frac{t^n}{[n]_q!}\,a_n^{(k)}(q), $$ where $[n]_q!$ denotes the $q$-factorial, admits an explicit closed form as a ratio of two $k$-periodically truncated $q$-exponential series. The proof is purely analytic and is based on a functional equation satisfied by $F_k(t;q)$, obtained via a decomposition of the $q$-exponential series into residue classes modulo $k$. Coefficient extraction yields a convolution identity involving Gaussian binomial coefficients, which uniquely determines the inversion enumerator. This provides an analytic alternative to classical inclusion–exclusion and structural combinatorial arguments for permutation classes with periodic descent constraints.

We develop a finite-kernel framework for sparse extremal graph counting. The problems considered here ask for the maximum number of copies or homomorphisms of a fixed graph under sparse edge constraints. In this regime, the leading term need not be governed by a single dense block. Instead, the extremal mass may be supported on several interacting asymptotic scales. Our framework identifies these scales via a finite-dimensional linear program, separates the leading contributions through a finite state decomposition, and synchronizes or realizes them inside a finite kernel. We apply this framework in three settings. First, we prove the sparse threshold conjecture of Day and Sarkar for graphons. For every fixed graph $H$ without isolated vertices, we prove that \[ \sup_{t(K_2,W)\le \beta} t(H,W)=\beta^{|V(H)|-\alpha^*(H)}(C_T(H)+o(1)) \] as $\beta\to0$, where $\alpha^*(H)$ is the fractional independence number of $H$ and $C_T(H)$ is an explicit sharp constant attained by a three-step threshold graphon. Second, we affirmatively answer a question of Blekherman and Patel by showing that, for every graph $H$, whenever $m\to\infty$ and $m=o(n^{3/2})$, threshold graphs asymptotically maximize $\hom(H,G)$ among all graphs with at most $n$ vertices and at most $m$ edges. Third, Gerbner, Nagy, Patkós, and Vizer conjectured that, among all bipartite graphs with $n$ vertices and $m$ edges, the quasi-complete bipartite graph asymptotically maximizes the number of copies of every fixed bipartite graph $H$ whenever $m=\omega(n)$ and $m\le n^2/4$. We disprove this conjecture in the subquadratic range and give the correct order of magnitude in terms of $\kappa_H(n,m)$, a finite-kernel scale defined by a finite-dimensional variational problem.

A family of spanning trees of a graph is $t$-intersecting if any pair of spanning trees in the family has $t$ or more edges in common. For sufficiently large $n$ and $t \leq n/C\log_2 n$ for some absolute constant $C>0$, we give a nearly complete characterization of the extremal $t$-intersecting families of spanning trees in balanced complete bipartite graphs with parts of order $n$. In particular, for $t=1$, we give exact bounds and a full characterization of the extremal families. For $t \geq 2$, our bounds are tight up to lower-order terms, and we show that any extremal $t$-intersecting family is of the form $\mathcal{F} \cup \mathcal{S}'$ where $\mathcal{F}$ is a family of all trees containing a fixed $t$-matching, and $\mathcal{S}'$ is a distinguished set of exceptional trees of size $|\mathcal{S}'| = o(|\mathcal{F}|)$.

For fixed $k\geq3$, we study the asymptotic number and typical structure of dense graphs with no induced copy of the star $K_{1,k}$. We solve the associated graphon variational problems both at fixed constant edge density $\gamma$ and for the conditioned Erdős–Rényi random graph $G(n,p)$ for constant $p$. As consequences, we obtain explicit formulas for the entropy density of induced-$K_{1,k}$-free graphs with $\Theta(n^2)$ edges and for the large deviation rate function for the event that $G(n,p)$ is induced-$K_{1,k}$-free. The entropy density exhibits a second-order phase transition at an explicit critical density $\gamma_k$, while the rate function exhibits a first-order phase transition at a critical parameter $p_k$. We completely characterize the optimizers of both variational problems. Both models have parameter values for which there are infinitely many optimal graphons, but there is always a unique graphon that represents the typical structure in cut metric. We refine the graphon-level results by giving a detailed structural description of both models. For supercritical parameters, each random graph model is the complement of a $(k-1)$-partite graph with high probability. In the subcritical regime of the fixed-density model, the typical structure is the disjoint union of the complement of a $(k-1)$-partite graph, and a sparse remainder. In the subcritical regime of the conditioned Erdős–Rényi random graph, a typical sample has $o(n^2)$ edges.

Significant combinatorial constraints and structure on matroids can come from notions in algebraic geometry, even without the matroids themselves being representable. Let \(M\) be a loopless matroid on a finite ground set \(E\), and let \(\G\) be a building set containing the top flat \(E\). We define a tangent class \(T_{M,\G}\) in the \(K\)-ring \(K(M,\G)\), which extends the tangent bundle class of the de Concini–Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class \(T_{M,\G}\) satisfies a matroidal Hirzebruch–Riemann–Roch package. More precisely, its Hirzebruch class specializes to the Todd class and computes the Chow polynomial of \((M,\G)\). In the realizable case, these identities agree with the usual tangent-bundle computations on the corresponding wonderful model. As an application, we prove Chern-number inequalities for \(T_{M,\G}\), including a Miyaoka–Yau type inequality with respect to the hyperplane class.

We prove an explicit spectral-gap lower bound for the lazy swap chain on binary matrices with prescribed row and column sums. This chain is a standard sampler for fixed-margin null models in ecology, statistics, and network analysis. Kannan, Tetali, and Vempala (KTV) conjectured that it mixes rapidly for all feasible margins \citep{kannan1997simple}. We show that for every feasible set of margins on an $m\times n$ binary matrix, the lazy swap chain has spectral gap at least $$\binom{m}{2}^{-1}\binom{n}{2}^{-1}.$$ The bound is tight in the worst case. Thus, our result proves this KTV conjecture in a stronger quantitative form. The same spectral-gap bound also verifies the Mihail–Vazirani conjecture for fixed-margin 0/1-matrix polytopes. The proof gives a new route to fixed-margin sampling that avoids stability assumptions and canonical-path constructions. We compare the swap chain with a two-row heat-bath chain and use a local-to-global spectral reduction to reduce the analysis from arbitrary $m\times n$ matrices to a three-row problem. The remaining three-row inequality is then proved by separating the scalar column-count sector from the non-scalar Johnson harmonic sectors. The proof itself was generated by ChatGPT 5.5 Pro. The author's role was to pose the problem, guide the search direction, evaluate the AI-generated arguments, rewrite the proof, and take responsibility for the final form and validity of the result. The full proof of the main theorem has been formalized in Lean, and the accompanying formalization is available at the anonymous repository https://github.com/guanyangwang/ktv-swap-lean.

The state space of an origami crease pattern forms a flip graph, whose vertices are the flat-foldable mountain-valley assignments and whose edges join assignments differing by a single face flip. For the $m \times n$ Miura-ori, the degree sequence and diameter of this graph are known only for two rows. Each assignment maps to an integer height function on the grid, under which a vertex's degree equals its number of local extrema. In this model the vertices of each degree up to five are counted by an explicit polynomial in $m$ and $n$, valid once both exceed a bound that grows with the degree, and the height functions realizing those degrees are described explicitly. A closed-form lower bound for the diameter holds for all $m$ and $n$, and the matching upper bound reduces to an extremal inequality for $1$-Lipschitz functions on the grid, recovering the two-row distance at $m=2$. Since each invariant is read from the extrema or height differences of a grid function, the same reduction applies to any flip-graph quantity expressible in those terms.

J.E. Littlewood posed the question of how many infinite circular cylinders of unit radius can be arranged so that each touches all the others. We give a computer-free proof that one cannot find ten such cylinders. This improves a known result, namely that there are no eleven such cylinders, which was obtained making partial use of computer verification.

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X, Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.

We extend the one-dimensional lonely set method to two dimensions for the purpose of studying the chromatic number of integer distance graphs in two dimensions. Given a lacunary sequence of displacement vectors in $Z^{2}$, we use a lacunary matrix theorem given by Broderick, Fishman and Kleinbock, to prove the existence of a satisfactory multiplier vector. We then give an explicit geometric colouring argument. This proves that any integer distance graph generated by a lacunary sequence of vectors in two dimensions has finite chromatic number.

We study edge ideals of sequentially Cohen-Macaulay co-chordal graphs through the maximal-clique structure of their chordal complements. After making the required construction order explicit, we derive a closed formula for the graded Betti numbers of a co-chordal graph whose complement is a $(d_1,\ldots,d_e)$-tree, and we characterize the Cohen-Macaulay case by its Betti sequence. The general formula is then specialized to split and threshold graphs, prime ideal graphs of finite rings, nilpotent graphs of products of finite chain rings, and zero-divisor graphs of finite chain rings. For products of chain rings, we characterize when the nilpotent graph is threshold and compute its Betti numbers in that range. Finally, among co-chordal zero-divisor graphs of $\mathbb{Z}_n$, we determine exactly when the quotient is sequentially Cohen-Macaulay: this occurs for $n=p^a$, $n=2p$, and $n=2p^2$, with $p$ odd in the last two cases.

A dominating set is called $k$-limited if every vertex in the set has at most $k$ neighbors outside it. The minimum cardinality among all $k$-limited dominating sets of a graph $G$ is the $k$-limited domination number, denoted by $\gamma_k^{\mathrm L}(G)$. We prove that, for every fixed integer $k\ge 2$, deciding whether a graph admits a $k$-limited dominating set of size at most $\ell$ is $\mathsf{NP}$-complete. In addition, a systematic study of $k$-limited domination in Cartesian products is initiated. In particular, we establish general lower and upper bounds for $\gamma_k^{\mathrm L}(G\square H)$, show that both are sharp, and derive exact values for several natural families of graph products.

In this work, we introduce and study the notion of $k$-limited domination in graphs, motivated by applications where dominating vertices have bounded capacity and cannot be overloaded by too many external neighbors. Formally, given an integer $k \le \Delta(G)$, a set of vertices $D \subseteq V(G)$ is called a $k$-limited dominating set if it is a dominating set and, in addition, each vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality of such a set is the $k$-limited domination number, denoted by $\gamma_k^{\mathrm{L}}(G)$. Since $\Delta(G)$-limited domination coincides with the classical domination, we restrict our attention to the nontrivial range $1 \le k < \Delta(G)$, where the degree limitation becomes meaningful and leads to new combinatorial phenomena. In this paper, we initiate the study of this concept by deriving sharp general bounds for $\gamma_k^{\mathrm{L}}(G)(G)$ and identifying conditions under which these bounds can be further improved. We establish a connection between $k$-limited domination and $(1,t)$-domination. In particular, for $d$-regular graphs we prove that $\gamma_k^{\mathrm{L}}(G)(G)=\gamma_{1,d-k}(G)$. In the special case $k=1$, we show that $1$-limited domination is tightly linked to graph packings, yielding the bound $\gamma^{\mathrm L}_1(G) \le n - \rho(G)$ and its characterization. The study reveals several natural open questions and indicates that limited domination provides a rich ground for further research.

A vertex coloring of a graph $G$ with nonempty color classes $V_1,V_2,\dots,V_k$ is called a \emph{FAT $k$-coloring} if there exist real numbers $\alpha,\beta\in[0,1]$ such that for every vertex $v$ and every color class $V_i \in \left\{ V_1,V_2,\dots,V_k \right\} $ we have $$ \bigl| N(v) \cap V_i \bigr|= \begin{cases} \alpha\deg(v) & \text{if } v\notin V_i,\\[4pt] \beta\deg(v) & \text{if } v\in V_i . \end{cases} $$ \noindent The FAT coloring concept was originally proposed and thoroughly studied by Beers and Mulas. The set of all FAT colorings of a graph is naturally ordered by the coarsening relation, in which finer partitions are larger in the order. The maximal elements of this poset, called irreducible FAT colorings, form a generating set: every FAT coloring of the graph can be obtained by merging color classes of some irreducible one. Beers and Mulas raised the compelling question whether, for every positive integer $s$, there exists a graph that admits exactly $s$ irreducible FAT colorings. In this paper we settle this question affirmatively by exhibiting, for any given $s$, a graph possessing precisely $s$ such colorings.

Panmagic permutations are permutations whose matrices are panmagic squares, better known as solutions to the modular n-queens problem, configurations of n non-attacking queens on a toroidal nxn chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets are a generalization of groups with N-ary multiplication instead of binary one. With the help of the Post coset theorem, we identify panmagic N-ary groups as cosets of the dihedral subgroup and its extensions in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory and combinatorics: square-free numbers, 4k+1 primes, quadratic residues, cycle indices from Polya counting, and linear congruential generators.

We investigate the minimum non-zero number of perfect matchings in planar graphs. We prove that this is a constant for 2-connected planar graphs of minimum degree 3 and 3-connected planar graphs. In the former case, the constant is 4 and this is best possible. In the 3-connected case, we give several infinite families, including nearly 3-regular graphs and triangulations with a constant number of perfect matchings. In contrast, it was known that in the 4-connected case the minimum non-zero number of perfect matchings is at least linear. For 5-connected triangulations, it follows from a result of Alahmadi, Aldred, and Thomassen that there must be exponentially many perfect matchings. The families of planar graphs we investigate here are classified by connectivity. We conclude the article with an infinite family of counterexamples to a conjecture published by Zaks; these are not planar, but very much concern connectivity constraints.

We initiate the study of transmission zero forcing, a variant of the well-studied zero forcing graph parameter. In this variant, a subset of vertices is assigned an initial unit weight, and these vertices can increase the weight of a neighbor subject to the zero forcing color change rule at a rate determined by the transmission proportion. A vertex is considered filled when its weight exceeds the transmission threshold, at which point the process can continue. The transmission zero forcing number of a graph is the minimum cardinality of the initial set that results in all vertices exceeding the transmission threshold. This iterative graph coloring process is a generalization of zero forcing that allows for a vertex to be forced by multiple neighbors. We develop tools for studying this graph parameter, determine its value on some common classes of graphs, and investigate its behavior under various graph operations.

Panmagic permutations are permutations whose matrices are panmagic squares. Positions of 1-s in the latter describe maximal configurations of non-attacking queens on a toroidal chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets have remarkable algebraic properties when one multiplies three or more of them rather than just two. In group-theoretic terms, they are special cosets of the dihedral group in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory: multiplicative orders, 4k+1 primes, primitive roots and quadratic residues.

The Hales-Jewett number HJ(t,r) is the least dimension n such that every r-coloring of the grid [t]^n contains a monochromatic combinatorial line. We prove HJ(3,3) >= 22 and HJ(4,2) >= 14, improving the previous records 14 and 12. The engine is an exact reduction: a coloring of [t]^n invariant under coordinate permutations descends to the discrete simplex of letter-count vectors, where a combinatorial line is precisely a corner tuple; the 4,387,586,157,901 lines of [3]^21 thereby compress to 1771 local conditions on 253 cells. We prove that this symmetric class coincides with the class of one-weight colorings, those reading an integer-weighted count of the letters: a radix weight realizes every symmetric coloring, so the symmetric lower-bound problem is a one-dimensional homothety-avoidance problem, the case d=1 of Gallai's theorem. This yields the closed-form bound HJ(t,r) >= ceil((G_r(S)-1)/D_S) in terms of the Gallai homothety numbers G_r(S), together with the new values G_3({0,1,3})=42, G_3({0,1,4})=57, G_2({0,2,3,5})=67, G_2({0,1,5,6})=80, and G_3({0,2,5})=77, giving HJ(3,3) >= 16 from a one-line certificate. Further results: periodic one-weight palettes give HJ^[12](3,3) = HJ^[12](4,2) = infinity for lines with at most twelve active coordinates; the interval number HJ^(1)(3) is exactly 5; a Rado reading gives G_4({0,1,3}) >= 94 and R_4(z+2x=3y) >= 59; and a rainbow companion gives the anti-Hales-Jewett bound ah(3,4) >= 25. A SAT program written for this article pushes the computation further: eighteen exact two-color Gallai numbers of four-point sets, up to G_2({0,1,6,7}) = G_2({0,3,4,7}) = 79; the exact three-color value G_3({0,1,5}) = 70; and the exact Rado numbers R_r(z+kx=(k+1)y) for 2 <= k <= 5 and r in {2,3}. Every displayed certificate is verified by direct enumeration; certificates and verification scripts are available at https://github.com/ysmouhib/hj-certificates.

Let $G$ be an edge-colored graph on $n$ vertices, and let $\deltac(G)$ denote its minimum color degree. Li and, independently Li, Ning, Xu, and Zhang, proved that every edge-colored graph on $n$ vertices with $\deltac(G) \ge \frac{n+1}{2}$ contains a rainbow triangle. Let $\rt(G)$ denote the number of rainbow triangles in $G$, and define \[ f(n) = \min\{ \rt(G) : |V(G)| = n,\ \deltac(G) \ge (n+1)/2 \}. \] In \cite{LiNingShiZhang2024}, the following open problem was posed: determine all the values of $f(n)$. In this paper, we determine $f(n)$ completely: $f(n) = (n^2-1)/8$ for odd $n\geq 3$, $f(n) = \frac{n^2}{4} - 1$ for all even $n \ge 6,$ and $f(4) = 4$. This resolves an open problem raised in \cite{LiNingShiZhang2024}.

In this note, we look at the structure and properties of palindromic or Pascal-like Sprugnoli arrays. We show that there are two closely related families of these arrays. We give closed form expressions for the elements of these families, and in each case, we describe the form of the inverse arrays. Finally, we consider the arrays modulo $2$ and the resulting arithmetic sequences.

For integers $n\ge d+1$, let $\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}. \] We introduce a two-cover lifting construction and prove the recursive lower bound \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+\mathsf{M}_{d-3}(n-5) \] for every $d\ge 3$ and $n\ge d+3$. Consequently, \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+\binom{n-6}{d-3}. \] Thus the Mubayi–Zhao conjecture on the exact value of $\mathsf{M}_d(n)$ for $n\ge2(d+2)$ is false for any $d\ge 3$. The proof is elementary and proceeds entirely through an explicit analysis of traces.

The recent work of Greaves, Jing, and Zhu gives an operator construction for the $t$-Schur functions and the $t$-Schur measure. Motivated by their construction, we consider the same type of vertex operator on the odd power-sum ring. Its Fourier modes generate a family of symmetric functions indexed by strict partitions, which we call shifted $t$-Schur functions. These functions specialize to Schur $Q$-functions at $t=0$. We derive a two-row formula, a Pfaffian Giambelli formula, a Cauchy identity, and a finite shifted Gessel-type formula. This note is intended as a first step toward further study of the odd-operator analogue of the Greaves–Jing–Zhu construction.

A graph $G$ is $H$-saturated if $G$ contains no $H$-copy as a subgraph, but adding any edge between two non-adjacent vertices in $G$ creates a copy of $H$. The saturation number $\mathrm{sat}(n,H)$ is the minimum number of edges in an $n$-vertex $H$-saturated graph. Cameron and Puleo [Discrete Math. 345 (2022), 112867] showed that $\mathrm{sat}(n,K_1\vee F)\le n-1+\mathrm{sat}(n-1,F)$ for all $n>|V(F)|$. It is natural to ask when the equality holds. Most known results on this bound consider non-empty graphs $F$ without isolated vertices. We study families in which $F$ contains isolated vertices. For $F=K_{p-1}\cup qK_1$, where $p\ge4$ and $q\ge p-1$, we prove that $\mathrm{sat}(n,K_1\vee F)=(p-2)n-\binom{p-1}{2}$ for $n\ge\max\{p+q,\binom p2\}$ and determine the unique extremal graph. We also give the exact result for $q=1$, where the extremal graph has a different structure. For $F=pK_2\cup qK_1$, in the natural range $n\ge2p+q+1$, we prove that $\mathrm{sat}(n,K_1\vee F)=n+3p-4$ holds exactly when $n\ge3p-2$, and the order condition is sharp. Furthermore, we characterize all extremal graphs.