Papers

Motivated by recent work on $\lambda$-fold near-factorizations by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate $(\lambda,\mu)$-fold factorizations as combinatorial structures with prescribed representation multiplicities. Such a factorization consists of two subsets of a finite group whose products represent each nonidentity element exactly $\lambda$ times and the identity exactly $\mu$ times. These structures generalize $\lambda$-fold near-factorizations and provide a common framework for several families arising in combinatorial design theory, including strong external difference families with two equal-sized subsets, as well as binary sequence pairs with two-level correlation. Their associated $\{0,1\}$-matrices satisfy the Bridges-Ryser equation $XY=\lambda(J-I)+\mu I$, linking the factorization problem to a classical matrix framework for combinatorial designs. We establish the foundational theory of $(\lambda,\mu)$-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.

The periodic tiling conjecture asserts that if a bounded set $\Omega\subset \mathbb R^d$ of finite measure tiles $\mathbb R^d$ by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in $\mathbb R$ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in $\mathbb Z^2$ was proved by Bhattacharya, whereas the conjecture in $\mathbb R^2$ remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments parallel to the axes. We prove that every such tiling that is topologically minimal is weakly periodic. Moreover, we show that if an axis-parallel polygonal set $\Omega$ tiles $\mathbb{R}^2$ then it admits a quasi-periodic tiling (i.e., a finite union of translated lattices). The proofs combine geometric decompositions of tilings with rational approximation and discrete tiling theory. As a consequence, we obtain that the translational tiling problem for axis-parallel polygonal sets with real algebraic vertex coordinates is decidable.

Minimum product-set problems ask how small a product of two subsets of prescribed cardinalities can be in a finite group. Although the problem is completely understood for finite abelian groups, exact results in the nonabelian setting are available only for particular classes. We determine the minimum product-set function $\mu_G$ for every finite metacyclic group, without assuming that the relevant cyclic extension splits. Our proof converts the extremal problem into a finite recursive arithmetic description of the feasible cardinality vectors of subset trios. The recursion combines the subgroup structure of finite metacyclic groups with the structural classification of maximal critical subset trios. The resulting characterization recovers the previously known formulas for abelian and nilpotent groups, dihedral groups, and nonabelian groups of order $pq$. It also yields metacyclic examples in which the minimum product-set size lies strictly between the classical lower and upper arithmetic bounds for finite solvable groups. For faithful split metacyclic groups, we further show that requiring one factor to be symmetric, contain the identity, and generate the whole group can force a strictly larger product set than the unrestricted minimum. The latter constraint admits a vertex-isoperimetric interpretation on connected Cayley graphs.

It is known that there can be at most 3 disjoint nontrivial minimal graphs in R^3 with 0 boundary values, and conjectured that there can be at most 2. We explore properties that the graphs would have if there were 3.

The bipartite edge frustration of a graph $G$, $\varphi(G)$, is the minimum number of edges that must be removed from $G$ to obtain a graph that is bipartite. Core graphs are singular graphs on $2$ or more vertices, for which the kernel of the adjacency matrix contains a full eigenvector. A nut graph is a core graph of nullity one. Nut graphs $G$ are non-bipartite. It is proved here that $\varphi(G) \geq 2$ if $G$ is a nut graph, i.e. every nut graph requires deletion of at least two edges to produce a bipartite graph. In addition, we give a constructive proof that for every integer $k \geq 2$ there exist infinitely many chemical nut graphs $G$ with $\varphi(G) = k$. Furthermore, for $3 \leq d \leq 8$, the values $k$, for which a $d$-regular nut graph $G$ with $\varphi(G) = k$ exists, are characterised. It is shown here that for every integer $k \geq 2$ there exist infinitely many cubic polyhedral nut graphs $G$ with $\varphi(G) = k$. Two ways of producing larger nut graphs from smaller are described: the cut and frustration constructions. One straightforward consequence is that total double subdivision of a nut graph (i.e. simultaneous insertion of two degree-$2$ vertices into every edge) yields a nut graph.

Let $C_n^k$ denote the $k$th power of the cycle $C_n$. The equivalence covering number $\operatorname{eq}(G)$, studied by Alon, is the minimum number of equivalence subgraphs needed to cover all edges of $G$. King conjectured that $\operatorname{eq}(C_n^k)=\Omega(k)$ for sufficiently large $n$ relative to $k$. We study $\operatorname{eq}(C_n^k)$ and prove that, for every $n \geq 2k + 2$, we have $1 + \lceil\log_2(k+1)\rceil \leq \operatorname{eq}(C_n^k) \leq 1 + 2\lceil\log_2(k+1)\rceil$. We further show that, whenever $n\ge(k+1)\max\{41,5\lceil\log_2(k+1)\rceil\}$, \begin{equation*} \operatorname{eq}(C_n^k)\in\left\{1+\lceil\log_2(k+1)\rceil,\,2+\lceil\log_2(k+1)\rceil\right\}. \end{equation*} Moreover, for every integer $n\ge \max\{2k+2,k^2(k+2)\}$, the following exact values hold in the indicated cases: \begin{equation*} \operatorname{eq}(C_n^k)=\begin{cases} 1+\lceil\log_2(k+1)\rceil, & k+1\neq2^{\lceil\log_2(k+1)\rceil} \text{ or }(k+1)\mid n,\\[3pt] 2+\lceil\log_2(k+1)\rceil, & k+1=2^{\lceil\log_2(k+1)\rceil} \text{ and }2\nmid n. \end{cases} \end{equation*}

Using the geometric realization spectral sequence and the Serre spectral sequence, we compute $K_1(\operatorname{Mat}_*)$, the first K-theory group of the proto-exact category of pointed matroids, answering a question of Jun, Sistko and Wright.

We give the topological classification of some simplest univariate complex fewnomials, and find tropical fans of the fewnomial strata with each topological type. Specifically, for the support $ A=\{d,d+1,d+2,d+3,d+4\},\quad d\geq3, $ we classify the topological equivalence classes of polynomials having two critical values of ramification type $(2,2)$. There is one class for odd $d$ and two classes for even $d$. We describe the branches at infinity of the corresponding one-dimensional stratum using Newton polytopes of certain complete intersection representations of the sought strata, and determine their tropicalizations. The combinatorial types of the obtained tropical polynomials distinguishes all three kinds of topological equivalence classes. These computations suggest a correspondence between topological equivalence classes of polynomials and combinatorial types of tropicalizations of their branches at infinity, as well as a possible periodicity phenomenon under translations of the support.

For a graph $G$, let $\nu(G)$ denote the maximum number of pairwise edge-disjoint triangles and let $\tau(G)$ denote the minimum number of edges meeting every triangle. Tuza's conjecture asserts that $\tau(G)\le 2\nu(G)$ for every graph $G$. We first note that Tuza's conjecture is equivalent to the following statement: for every fixed $c\ge 0$, if $G$ is a sufficiently large $n$-vertex graph with $e(G)\ge n^2/4-cn$, then $\tau(G)\le 2\nu(G)$. In this paper, we prove this statement for every $c<2$.

We study the variation of hull dimensions of symplectic linear codes under equivalence transformations preserving the symplectic distance. For a symplectic linear code $C$, we introduce an even-valued invariant $\Delta(C)$, which depends only on the equivalence class of $C$. This invariant can be expressed as the generic rank of an associated space of alternating matrices and, equivalently, as twice a corresponding matroid matching number. If $C$ has dimension $k$, we prove that the minimum symplectic hull dimension among all codes equivalent to $C$ is $k-\Delta(C)$. Consequently, $C$ is equivalent to a symplectic LCD code if and only if $\Delta(C)=k$. We further show that every symplectic MDS code of even dimension is monomially equivalent to a symplectic LCD code. For a symplectic self-orthogonal code $C$, we determine all attainable hull dimensions: for every integer $r$ with $0\le r\le \Delta(C)/2$, there exists a monomially equivalent code whose symplectic hull has dimension $k-2r$. We also refine a standard form for the generator matrix of a symplectic self-orthogonal code and derive the corresponding parity-check matrix and the Gram matrix of the dual code. In the self-dual case, we characterize $\Delta(C)$ through minors of an associated symmetric matrix. Furthermore, we relate hull variation to graph matchings, establish a bound for $\Delta(C)$, and give graph constructions producing symplectic self-dual MDS, optimal, and best-known codes.

A connected graph $G$ is matching covered if it contains at least one edge and every edge belongs to some perfect matching of $G$. A matching covered graph is a brick if it is 3-connected and after removing any two vertices, the resulting graph has a perfect matching. Bricks are the building blocks of matching covered graphs. A brick $G$ is extremal if the incidence vectors of all perfect matchings are linearly independent over $\mathbb{R}$. Zhou et al. [On planar extremal nonsolid bricks, J. Graph Theory, 2026] proved that for every planar extremal brick $G$, there exists a sequence of planar extremal bricks $G_0,G_1,\dots,G_n$ such that the underlying graph of $G_n$ is an odd wheel, $G_0=G$, and each $G_i$ $(0\leq i < n)$ has a triangle $T_i$ such that $G_i/T_i = G_{i+1}$, where $G_i/T_i$ means contracting the triangle $T_i$ into a single vertex. In this paper, the result of Zhou et al. is generalized to planar PM-compact bricks, a larger class of bricks including extremal bricks. We present a chain theorem for planar PM-compact bricks: for every planar PM-compact brick $G$, there exists a sequence of planar PM-compact bricks $G_0,G_1,\dots,G_n$ such that the underlying graph of $G_n$ is an odd wheel, $G_0=G$, and each $G_i$ $(0\leq i < n)$ contains a triangle $T_i$ such that $G_i/T_i = G_{i+1}$.

FirstFit colors each arriving vertex with the smallest positive integer absent from its previously colored neighborhood. We prove that this rule can require $n^{\Omega(1)}$ many colors on certain bipartite graph of $n$ vertices even when the vertices arrive in a uniformly random order. This refutes the conjecture of Bosek, Gutowski, Lasoń, and Przybyło [BGLP24] that FirstFit uses a polylogarithmic expected number of colors on every bipartite graph in the random order model. We obtain the lower bound by analyzing the FirstFit coloring process on random bipartite graphs, where vertices arrive to either side with equal probability, and with an independent edge density $n^{-x}$ for some $0 < x < 1$. Our result can also be viewed as a qualitative strengthening of Kučera [Kuč91], who showed that, for every $\epsilon > 0$, there are graphs with chromatic number at most $n^{\epsilon}$ on which FirstFit uses $\Omega(n/\log n)$ colors in expectation under a uniformly random order. Our result shows that such poor behavior of FirstFit already occurs for $\chi(G) = 2$.

For an $r$-uniform hypergraph $\mathcal H$, let $\ex_r(n,\mathcal H)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no copy of $\mathcal H$. The $r$-uniform expansion $T^{(r)}$ of a tree $T$ is obtained by adding $r-2$ new vertices to each edge, with the added sets pairwise disjoint and disjoint from $V(T)$. In particular, the expansion of $K_{1,k}$ is the linear star $\mathcal L_k^r$, whose $k$ edges intersect pairwise in a single common vertex. Chung and Frankl (1987) determined $\ex_3(n,\mathcal L_k^3)$ and all extremal hypergraphs for every $k\ge3$ and sufficiently large $n$. We extend the odd-$k$ part of their result to every $r\ge5$, determining the exact Turán number and the unique extremal hypergraph up to isomorphism for sufficiently large $n$. More generally, we extend this result to expansions of trees for which the minimum degree among the vertices in the smaller color class is odd and at least three. For every $r\ge5$ and sufficiently large $n$, we establish lower and upper bounds on their Turán numbers. When no two vertices attaining this minimum degree in that color class have a common neighbor, we determine the exact Turán number and the unique extremal hypergraph up to isomorphism. The proofs combine the delta-system method with stability arguments and a detailed analysis of extremal constructions.

Let $f(n)$ be the largest integer such that every set of $n$ points in the plane contains $f(n)$ points no two at distance $1$, and let $m_1$ be the supremum of the upper densities of Lebesgue measurable planar sets with no two points at distance $1$. Larman and Rogers observed that $f(n) \ge m_1 n$. We prove that $f(n)/n \to m_1$; that is, $f(n) = (m_1 + o(1))n$. This is the first half of a conjecture of Dúcz and Varga, and with the bound $m_1 \le 0.247$ of Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki it gives $f(n) \le (0.247 + o(1))n$. The proof builds a random independent subset of the algebraic plane, invariant under algebraic translations and rotations, whose probability of containing the origin is at least the worst finite independence ratio, and turns it into a measurable planar set of at least that upper density. Its only non-classical input is the Haar rigidity theorem for wild character laws from the OpenAI preprint The Euclidean plane is not five-colorable, which has a machine-checked Lean proof. The whole argument has been formalized in Lean 4 on top of it.

Suppose that each of $n$ players rolls a different die and the players are ordered by their rolls. Can the dice use disjoint labels such that ties are impossible, while all $n!$ orders remain equally likely? We give two general constructions of such dice and solve some related problems.

Let $h\geq2$ be an integer and let $\mathbb{N}_0=W_1\cup\cdots\cup W_h$ be a partition into infinite sets. For $W\subseteq\mathbb{N}_0$, let $A(W)$ denote the set of all finite nonempty sums of distinct powers $2^w$ with $w\in W$. We give a necessary and sufficient condition for $A=A(W_1)\cup\cdots\cup A(W_h)$ to be a strongly minimal asymptotic basis of order $h$. This provides a complete answer to a problem posed by Nathanson in 1988.

Let $\mathcal{C}_{r}^+$ be the set of all balanced signed graphs with underlying graphs $C_r$. A signed graph is called $\mathcal{C}_{r}^+$-free if it contains no subgraph isomorphic to any member of $\mathcal{C}_{r}^+$. In this paper, we characterize the extremal graphs that achieve the maximum index among all $\mathcal{C}_{2k+1}^+$-free unbalanced signed graphs of order $n$ when $k\ge 2$ and $n\ge 187k$.

We exhibit a fixed $4\times 4$ binary matrix with no polynomial induced ordered removal bound. At depth $h\geq 1$, let $m=2^h$. Our explicit host has order $(14h+2)m$, exactly $2m^6+\frac{1}{2}(m-2)m^5$ induced copies, and exact editing distance $m^2$ cells. Its normalized editing distance is $\Theta(h^{-2})$, while its induced-copy density is $\Theta(4^{-h}h^{-8})$. The lower bound holds against arbitrary cell edits through an injective family of $m^2$ protected selections. An exhaustive arithmetic classification identifies every induced copy and proves that a matching canonical repair creates none. The original-copy transversal number is exactly $m$, yielding an unbounded gap between hitting existing copies and eliminating all copies. The identities and the obstruction at every depth are formalized in Lean, with verification sources included as ancillary files.

For a $k$-uniform hypergraph $L$, let $h_L(n,q)$ be the minimum number of copies of $L$ in an $n$-vertex $k$-graph with $\mathrm{ex}(n,L)+q$ edges. Let $t_L(n,q)$ be the minimum number of copies obtained by adding $q$ edges to an extremal $L$-free $k$-graph. A graph is color-critical if deleting a suitable edge decreases its chromatic number. Let $\mathcal K_k(F)$ be the expanded graph obtained by adding $k-2$ new vertices to each edge of $F$, with disjoint sets of new vertices for distinct edges. Mubayi proved that, for every clique $K_r$ with $r\ge4$ and fixed $ q>0$, $ h_{\mathcal K_3(K_r)}(n,q)\ge qt_{\mathcal K_3(K_r)}(n,1)$ for every sufficiently large $n$. Pikhurko and Yilma later proved that, for every color-critical graph $F$, there is a constant $\epsilon_F>0$ such that $h_{F}(n,q)=t_{F}(n,q)$ for every sufficiently large $n$ and $1\le q\le\epsilon_F n$. We extend this equality to expanded color-critical graphs. Let $d>k\ge3$. For every color-critical graph $F$ with $\chi(F)=d+1$, there is a constant $\epsilon>0$ such that $ h_{\mathcal K_k(F)}(n,q)=t_{\mathcal K_k(F)}(n,q)$ for every sufficiently large $n$ and $1\le q\le\epsilon n^{k-1}$. Let $q_L(n)$ be the largest integer $q$ such that $h_L(n,s)=t_L(n,s)$ for every integer $1\le s\le q$. We give a sufficient condition for $q_L(n)=\Theta_L(n^{k-1})$ and an explicit family satisfying it. We also give a different sufficient condition under which $q_L(n)=\Omega_L(n^{k-1/2})$. \noindent Keywords. Supersaturation; extremal graph; color-critical graph; chromatic number; expanded graph; hypergraph.

We introduce and study trigonometric weight functions of type $B$ and type $C$ arising from integrable lattice models. These functions give $q$-deformations of the factorial $K$-theoretic Schur $P$- and $Q$-functions of Ikeda and Naruse, with the $P$-functions specialized to the type $B$ case. We realize them as specializations of partition functions of a six-vertex model with triangular boundary conditions and derive their algebraic properties using the Yang-Baxter equation and properties of the $R$- and $K$-matrices. We introduce normalized weight functions and establish diagonal evaluation formulas, divisibility properties, and bounds on Newton polytopes analogous to those in type $A$, together with recursive relations and stability. We further derive branching formulas relating these functions to type $A$ weight functions, and prove equal-cardinality orthogonality and all-cardinality biorthogonality relations.

We characterize non-commutative polynomials of degree 2 which preserve free infinite divisibility as self-adjoint Lie polynomials of degree 2. This extends previous results which characterized homogeneous quadratic forms only. The proof is based on an involution of colored noncrossing partitions cancelling many terms; positive definiteness of the remaining terms is established by an explicit operator model on Fock space. Examples show that higher order Lie polynomials do not preserve free infinite divisibility.

In this short note, we point out that the recent lower bounds for off-diagonal Ramsey numbers $r(s,t)$ with fixed $s\ge 5$, together with the recent proof of the Ajtai–Erdős–Komlós–Szemerédi independence conjecture, imply \[ r(4,t)=\frac{t^3}{(\log t)^{2+o(1)}}. \] This note is solely for informal circulation and will not be submitted for publication.

Let $\Lambda_m$ be the Barnes-Wall lattice of dimension $2^m$, normalized to have minimum squared norm $d_m:=2^{m-1}$. We determine its complete norm spectrum strictly between $d_m$ and $2d_m$, proving that, for every integer $m\ge1$, $\{\|x\|^2:x\in\Lambda_m,\ d_m<\|x\|^2<2d_m\}=\{2d_m-2^j:j\in\mathbb Z,\ \lceil m/2\rceil\le j\le m-2\}$. In particular, the norm spectrum in this interval is nonempty for $m\ge4$. The proof uses a Gaussian-integer model in which lattice automorphisms arising from Clifford unitaries transform a vector into a pair of coordinate subvectors in a lower-dimensional lattice. Divisibility of the differences of their squared norms and a matrix Parseval identity provide the arithmetic and averaging constraints needed for induction on $m$. Every listed norm is realized explicitly by a sum of two minimal vectors of the lattice.

For a finite simple graph $G$, let $\nu(G)$ be the maximum number of pairwise edge disjoint triangles and $\tau(G)$ the minimum number of edges meeting every triangle. Tuza conjectured in 1981 that every such $G$ satisfies $\tau(G) \le 2 \nu(G).$ We prove the conjecture for the class of graphs with $\Delta\le10.$

To bridge the gap between worst-case and heuristic analyses of lattice enumeration, we study the sizes of (unpruned) enumeration trees on $n$-dimensional Haar-random lattices. Writing $n^{cn+o(n)}$ for the tree size at the shortest vector radius, we investigate the asymptotic scaling of $c$, and which bases minimize it. For quasi-HKZ bases, we rigorously analyze the random-lattice HKZ geometry and establish $c=1/(2e)\approx 0.1839$. This matches the worst-case enumeration exponent of Hanrot-Stehlé and confirms earlier predictions for the geometry of HKZ bases on random lattices. Searching beyond quasi-HKZ bases, we prove that with high probability, the minimum achievable tree size exponent $c$ over all bases of a random lattice satisfies: \begin{align} 0.1464 \approx \frac{2-\sqrt{2}}{4}\leq c<0.1474. \end{align} For the lower bound, we use Kim's mean-value formula for nested chains of primitive sublattices to rule out bases with small prefix determinants. This strengthens the earlier Rankin-based heuristic lower bound exponent $c \geq 1/8$, discussed by Nguyen and Hanrot-Stehlé, and rules out the existence of random-lattice bases with trees as small as predicted by the geometric series assumption. For the upper bound, we show that with high probability a dense primitive half-rank sublattice exists with determinant at most $n^{n/8 + o(n)}$. Combining dual HKZ on this sublattice with HKZ on its quotient gives a basis with $c \leq 1/(4\sqrt{e}) \approx 0.1516$. Recursively splitting each half lowers the upper bound below $0.1474$, and under a natural heuristic assumption we prove that the lower bound is tight. Combined with the collision search of Feng-Gao-Liu, our two-block construction gives a polynomial-space algorithm solving the closest vector problem with preprocessing (CVPP) on random lattices in query time $n^{0.0892 n + o(n)}$ for arbitrary targets.

Consider an $n$-dimensional sum of the form $S_k=\sum{q^{f({\bf z})}}$ over lattice points ${\bf z}$ which lie in $K{\mathcal P}$, the $K$ scaling of an $n$-dimensional rational polytope $\mathcal P$. We formulate a conjecture on the structure of $S_K$ considered for $K$ prime, as a power series in $h=q-1$ modulo the $K^{th}$ cyclotomic polynomial, and in particular the existence of a decomposition in powers of the Gauss sum $\mathbb G$, up to the $n^{th}$, whose coefficients are almost independent of $K$, where the leading coefficient is related to the volume of $\mathcal P$ and the coefficient of ${\mathbb G}^r$ has contributions from the codimension $r$ skeleton of $\mathcal P$. The conjecture is proved for $n=1,2$.

For integers $k\ge3$ and $0\le \ell<k$, we call a graph $(\ell\bmod k)$-cycle-restricted if every cycle has length congruent to $\ell$ modulo $k$. Akbari, Jamaali, Mahmoody, and Seyed Fakhari [On the size of graphs whose cycles have length divisible by a fixed integer, Australas. J. Combin. 45 (2009), 67–72] determined the maximum number of edges in a $2$-connected $(0\bmod k)$-cycle-restricted graph and characterized the graphs attaining equality. In this paper, we determine the maximum number of edges in a $2$-connected $(\ell\bmod k)$-cycle-restricted graph for every $\ell \neq 0$, and completely characterize all graphs attaining equality. Combined with the result of Akbari et al., our result gives a complete solution for all residue classes modulo $k$.

In a previous work, we defined a quotient graph of the Bruhat-Tits tree associated to the degree two special linear group over a local field whose vertex set is in correspondence with the maximal orders in a certain number of spinor genera. This graph has been applied to the study of the structure of some arithmetic groups, their presentations and their (co)homology groups. The genera of Eichler orders, which, for instance, are related to the problem of finding a basis that simultaneously splits two vector bundles with the same generic fiber, can often be described in terms of simpler genera via quotients. Furthermore, some embedding problems of orders can be studied via quotient graphs. These results provide a geometric framework for the study of maximal orders over function fields. In this article, we extend that construction to a higher dimensional setting through the use of singular chain complexes. Then we describe the set of vertices in this quotient that correspond to split maximal orders and the topology of the corresponding region of the quotient complex. More precisely, we show that these “split vertices” fall into a finite number of quotients of apartments by suitable subgroups of the corresponding Weyl group, the symmetric group. We relate these “quotient apartments” with the cuspidal sectors defined by previous authors who studied the quotient by the group of integral points of the special linear group. Both quotients are indeed related by a ramified cover that is unramified on cusps, as proved here.

The Game of Cycles, invented by Francis Su, is a two-player game in which the players take turns drawing arrows on the edges of a plane graph, never creating a sink or a source. On a tree only the sink-source rule matters, and the player who makes the last move wins. Several authors conjectured that on a tree the second player wins exactly when the number of markable edges is even. We show that this parity conjecture is false, and we explain why. The explanation is a hidden game of Nim: a path segment that is blocked by an arrow at one end and free at the other behaves exactly like a nim heap whose size is its length. Combined with a calculus for cutting and gluing trees at vertices next to leaves, this shows that the conjecture fails as soon as a single tree has Grundy value at least 2. The smallest counterexamples have 15 vertices. Trees with Grundy values 2 and 4 first appear at 17 vertices, and every value up to 63 occurs, which answers a question of Fokkink and Zandee. We also refute a related conjecture of Zandee. We then look closely at spiders. The spiders S(2,3,z) violate the conjecture for every z >= 14. The Grundy values of S(1,2j,z) are given by an exact formula governed by binary carries. For 99 pairs (x,y), the value of S(x,y,z) is eventually the parity of z. On the positive side, the conjecture holds for all trees in which every thread has at most two edges, and this bound is sharp. Along the way we prove a twin-branch reduction, give an algorithm that is polynomial for spiders with a fixed number of legs, and collect the questions that we could not answer. The boldest asks whether every non-negative integer is the Grundy value of some tree.

Chromatic choosability is a fundamental notion in list coloring. A graph $G$ is chromatic-choosable if its chromatic number $\chi(G)$ equals its list chromatic number $\chi_{\ell}(G)$. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted by $P_{\ell}(G,m)$, which is the list-coloring analogue of its chromatic polynomial $P(G,m)$. A graph $G$ is enumeratively chromatic-choosable if $P_{\ell}(G,m)=P(G,m)$ for every $m \in \mathbb{N}$. In 2002, Ohba proved that for every graph $G$, there exists an $N \in \mathbb{N}$ such that $K_p \vee G$ is chromatic-choosable whenever $p \geq N$. Allred and Mudrock recently conjectured that an analogous statement holds for enumerative chromatic choosability. In this paper, we prove that $K_1 \vee G$ is enumeratively chromatic-choosable whenever $G$ is a theta graph. Our proof combines classical inequalities, the polynomial method, and ideas from DP (or correspondence) coloring.