Papers

We establish weighted bilinear summation identities for two families of Apéry-like polynomials $g_n(x)$ and $v_n(x)$. The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For $g_n(x)^2$ we obtain identities with weights $(2n+1)^r$ for $1\le r\le4$; for $v_n(x)^2$ we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo $p^3$ and $p^4$, together with special evaluations modulo $p^5$ and $p^7$, where $p$ is a prime greater than $3$. In particular, $$\sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2 \equiv 6p^4-\frac{143}{3}p^6\pmod {p^7},$$ confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and $p$-adic endpoint expansions.

This text is a self-contained expository survey of the Lagrange and Markov spectra, centered on a comprehensive exposition of Markov's theorem and its generalizations. Its purpose is to provide a systematic text for learning the theory, with detailed proofs and explanations of the connections among its arithmetic, combinatorial, and geometric descriptions. The necessary background in continued fractions, quadratic irrationals, binary quadratic forms, and bi-infinite sequences is developed step by step, followed by an exposition of generalized Markov numbers, fence posets, curve lengths, and generalized Cohn matrices. One goal of this exposition is to explain the formula connecting generalized Markov numbers with the two spectra. For nonnegative integer parameters $(k_1,k_2,k_3)$, a permutation $\sigma\in\mathfrak S_3$, and a fraction label $t\in\mathbb Q_{\geq0}\cup\{\infty\}$, let $m_t$ be the associated generalized Markov number and let $k_t=k_{i_t}$ be the parameter assigned to its position $i_t\in\{1,2,3\}$. The text explains the construction of an associated finite sequence $S(t)$ of positive integers and the identity \[ \mathcal L(\alpha_{S(t)}) =\mathcal M(Q_{S(t)}) =\frac{\sqrt{((3+k_1+k_2+k_3)m_t-k_t)^2-4}}{m_t}, \] where $\alpha_{S(t)}=[\overline{S(t)}]$ and $Q_{S(t)}=(x-\alpha_{S(t)}y)(x-\alpha'_{S(t)}y)$, with the prime denoting quadratic conjugation. Here $\mathcal L$ and $\mathcal M$ denote the Lagrange and Markov constants, respectively. The survey explains how this identity relates generalized discrete Markov spectra to the classical theory and how Markov's theorem is recovered when the parameters vanish. The account also includes boundary values arising from irrational slopes and generalizations of Frobenius's uniqueness conjecture, providing a unified perspective on the classical theorem and its extensions.

We provide a general upper bound on multicolor ordered Ramsey numbers in terms of the interval chromatic number and the degeneracy of an ordered graph. We extend previous results by Conlon, Fox, Lee, and Sudakov (2017) by showing that for every $n$-vertex ordered graph $G^<$ with degeneracy $d\geq2$, and interval chromatic number $\chi$, its $q$-color ordered Ramsey number satisfies $r_<(G^<;q) \in n^{O(d^{q-1}\lceil \log{\chi}\rceil^{q-1})}$ for every $q \geq 2$. For fixed parameters $q,d,\chi$, the resulting estimate is polynomial in $n$. For triangle-free ordered graphs $G^<$, we also provide the stronger estimate $n^{O(q^2d {\lceil \log \chi \rceil}^{q-1})}$. It also follows from a recent result by Li (2026) that our upper bound is almost tight for ordered matchings.

In this paper, we study the Abelian maximal pattern complexity $p_{\alpha}^{\ast \mathrm{ab}}(k)$, introduced by Kamae, Widmer and Zamboni, of infinite words $\alpha \in \mathbb{A}^{\mathbb{N}_{0}}$ over finite alphabets $\mathbb{A}$. For recurrent aperiodic words, we determine a lower bound and prove its sharpness. We further give a structural characterization of the words with minimal Abelian maximal pattern complexity. In the general case, we prove that an infinite word $\alpha $ is aperiodic if and only if $\binom{p_{\alpha }^{\ast \mathrm{ab}}(k)}{2}\geq k$ for every $k.$ For aperiodic words over $\ell \geq 2$ letters, each occurring infinitely often, we further prove that $p_{\alpha }^{\ast \mathrm{ab}}(k)\geq m$ whenever $\binom{m}{2}\leq (\ell -1)(k-\ell +2)$, for all $m,k$. Together with a matching construction, this shows that the minimum Abelian maximal pattern complexity in this class is $\sqrt{2(\ell -1)k}+O_{\ell }(1)$. We call a word an Abelian pattern Sturmian word if, at every $k$, its Abelian maximal pattern complexity is the least positive integer $m$ satisfying $\binom{m}{2}\geq k$. We show that a word is Abelian pattern Sturmian if and only if, after relabeling its alphabet, it is the characteristic word of an infinite set $E\subset \mathbb{N}_{0}$ for which the bipartite graph on two disjoint copies of $\mathbb{N}_{0}$, with a left vertex $r$ adjacent to a right vertex $s$ exactly when $r+s\in E$, is a forest.

This paper has two aims. First, we study nowhere-zero flows in countably infinite graphs, observing the different roles played by two types of bridge. Second, we ask how typical graphs with such bridges are. Since there is no canonical probability measure on countable graphs, we explore an analytic approach and study which graph classes are nowhere dense and which are meager for two different definitions of a distance. We characterize in this sense graphs with a bridge (separating a finite component, or two infinite ones), $D$-edge-colorable graphs, etc.

Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of their presentation in the context of the coordinate rings of regular nilpotent Hessenberg varieties. In particular, they connect the coordinate ring of the lower unipotent group in a general linear group and the polynomial ring with quantum parameters. In this paper we consider the dual of the connection and we see that the Plücker coordinates correspond to the quantizations of Schur polynomials. As an application of the connection, we construct an action of the symmetric group on the polynomial ring with quantum parameters. Using the symmetric group action, one can define the divided difference operators on the polynomial ring with quantum parameters. We study the quantizations of Schubert polynomials in relation to the divided difference operators.

In an $n$-dimensional cropped hypercube each of the $2^n$ cropping inequalities chops off a single corner of the $0$–$1$ hypercube by an $\ell_1$-distance $\rho$. The case $\rho = 1/2$ has been extensively studied in the literature. This paper shows that the Lasserre rank of the $n$-dimensional cropped hypercube where $\rho = 1/2$, $n \geq 2$, is the smallest integer $0\leq t \leq n$ such that $\Delta_t < 0$ in the recurrence $\Delta_{-1} = 1$, $\Delta_{0} = n-1$, $\Delta_t = (n-1)\Delta_{t-1} - t(n-t+1)\Delta_{t-2}$. It follows that the Lasserre rank can be computed in time $O(n^2 \log^2 n)$. Asymptotically, the rank is $\frac{n}{2} + c_{1/2}\sqrt{n} + o(\sqrt{n})$, where $c_{1/2}$ is the unique zero of a given function. Numerically, $c_{1/2} \approx 0.3825$. In fact, we prove such results for any fixed $0 < \rho < 1$.

We study lattice-counting functions on compact integral-integral affine manifolds with corners, which generalize integral unimodular polytopes. We prove analogues of Ehrhart's theorem and Ehrhart-Macdonald reciprocity in this setting.

A subset $S \subseteq \mathbb{F}_2^n$ is called a Sidon set if no four distinct points of $S$ have zero sum. It is shown that if $n \geq 7$ and $n \equiv 3 \mod 4$, then $|S|\leq 2^{\frac{n+1}{2}}-3$. As a consequence, for any even $t \geq 4$, there does not exist a binary linear $[2^t-3,2^t-2t-2,5]$-code, strengthening a nonexistence result of Brouwer and Tolhuizen from 1993. As a second consequence, we also show that for an even integer $n \geq 4$, every almost perfect nonlinear (APN) function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ has nonlinearity at least $3$.

In this paper, we prove that every connected sesqui-regular graph with parameters $(n,k,c)$, where $c\geq3$, and with smallest eigenvalue in $[-3,-2)$ is $1$-integrable if its valency is sufficiently large. The bound on $c$ is optimal: for every integer $t\geq2$, the Cartesian product of the Shrikhande graph and $K_t$ is a connected sesqui-regular graph with parameters $(16t,t+5,2)$ and smallest eigenvalue $-3$, but it is not $1$-integrable.

We establish a Hamiltonian-to-pancyclic analogue of Bondy's theorem for graph systems under an aggregate degree condition. Let $\G=(G_1,\ldots,G_n)$ be a graph system on a common $n$-vertex set $V$, and write $\delta(v)=\min_{i\in[n]}d_{G_i}(v)$. If $\G$ contains a rainbow Hamilton cycle and \[ \sum_{v\in V}\delta(v)\ge \left\lceil\frac{n^2}{2}\right\rceil-1, \] then $\G$ is rainbow pancyclic, unless $n$ is even and every member is the same balanced complete bipartite graph. For even $n$ the threshold is exact at the integer level. Unlike the usual transversal Dirac- or Ore-type hypotheses, our condition is not layerwise: the member attaining $\delta(v)$ may depend on $v$, and some vertices may have $\delta(v)<n/2$. Relative to a fixed rainbow Hamilton cycle, we count shortcuts whose colors are released by the Hamilton arcs they replace. A missing cycle length forces complementary shortcut supports to cross-intersect. A counting gap settles even shortening, while equality or near equality in odd shortening yields a distance-two exchange whose orbits force the balanced bipartite obstruction. At the lower integer threshold an exact defect identity shows that only one or two units of slack are available. \noindentKeywords: graph system; rainbow cycle; pancyclicity; Hamilton cycle; extremal graph theory.

We prove that there exists $A\subseteq \mathbb{N}$ such that \[ R_A(N)=\frac{\pi}{4}N+ O\!\left(N^{1/4}\sqrt{\log N}\right), \] where $R_A(N)=\#\{(a,b)\in A^2:a+b\le N\}$. This improves the record $O(N^{1/4}\log N)$ obtained by Ruzsa in 1997.

Duke's interpolation theorem states that the orientable genera of a connected graph form an integer interval, and Stahl established the corresponding result for nonorientable embeddings. In 1991, Širáň showed that this interpolation property fails for signed graph embeddings: the Euler-genus spectrum of a signed graph may contain gaps. He subsequently asked whether all such gaps must occur at the lower end of the spectrum. In this paper, we establish a characterization of the Euler-genus spectrum of a connected signed graph. We prove that, for each parity class, the Euler genera form a step-two interval. Moreover, whenever both parity classes are nonempty, their maximum elements differ by one. As an consequence, if two consecutive integers $k$ and $k+1$ belong to the Euler-genus spectrum, then every integer from $k$ to the maximum Euler-genus also belongs to the spectrum, thereby answering Širáň's question affirmatively. Our proof uses the pre-signed graph representation of signed embeddings together with ordered adjacent-exchange operations and a matching interpretation of face numbers.

We determine, up to isomorphism of the covering graphs, the connected arc-transitive regular covers of cubic edge-primitive graphs whose covering transformation group is cyclic or elementary abelian of order $p^2$, where $p$ is a prime. Combining the known classifications for the base graphs ${\rm K_{3,3}}$ and ${\rm DC_{14}}$ with new arguments for ${\rm F30A}$ and ${\rm F102A}$ gives the full list in these two classes of covering groups. In the cyclic case, the covers of ${\rm F30A}$ and ${\rm F102A}$ are ${\rm F90A}$ and ${\rm F204A}$, respectively. In the elementary abelian case, neither ${\rm F30A}$ nor ${\rm F102A}$ admits an arc-transitive regular $\mathbb{Z}_p^2$-cover, so the base graph is ${\rm K_{3,3}}$ or ${\rm DC_{14}}$.

Liftings and parallel redrawings are classical topics in applied discrete geometry, concerned respectively with vertically lifting a $(d-1)$-picture – a realisation of a vertex-hyperplane incidence geometry in $\mathbb{R}^{d-1}$ – to a $d$-dimensional polyhedral scene, and with redrawing a $d$-picture (or hyperplane arrangement) within $\mathbb{R}^d$ while preserving prescribed hyperplane normals. In this paper, we develop a unified framework for the analysis of forced-symmetric liftings and forced-symmetric parallel redrawings. In particular, we show that the classical duality for these theories also appears in the forced-symmetric setting. We establish the orbit lifting matrix and the orbit concurrence geometry matrix, and show that they are the appropriate symmetry-adapted analogues of the standard lifting matrix and concurrence geometry (or parallel redrawing) matrix. We also make explicit the relationship with the orbit version of Whiteley's parallel design matrix for graphs. Using these tools, we derive necessary conditions for symmetric pictures to be forced-symmetric flat, and for symmetric hyperplane arrangements to be forced-symmetric robust, expressed as sparsity counts on the group-labelled quotient graphs associated with the symmetric incidence geometries. Finally, we discuss conjectures regarding the sufficiency of these conditions for generic configurations.

A de Bruijn sequence packs all $n$-bit binary words into a cycle of length $2^n$. A de Bruijn torus is the two-dimensional analogue in which each word appears exactly once in a rectangular window. Here we consider the natural analogue for permutations using their shorthand representation (i.e., each permutation's final redundant value is omitted from the window). We show that these tori exist when $n = 2m + 1$ is odd and the torus and windows have two rows (i.e., the torus is a "tworus"). These twori can be constructed with a high degree of symmetry. More specifically, there are twori that can be partitioned into $2^{m-1}$ matching blocks where each block contains the same sequence of unordered columns. Furthermore, given one such block we can generate each successive column of a tworus in amortized $\mathcal{O}(1)$-time. We also prove non-existence results for certain sizes of tori and provide algorithms for constructing multiversal cycles (perfect necklaces) of unlabeled binary words.

Let $\Ga$ be a connected prime-valent $X$-arc-transitive Cayley graph of a finite characteristically simple group $G\cong T^k$, where $k\geqslant2$. We obtain a precise structural characterization of such graphs and their arc-transitive automorphism groups. In the cubic case, every connected symmetric Cayley graph of $T^k$, where $T$ is a finite nonabelian simple group, is normal.

The Tutte polynomial is a bivariate polynomial that has been extensively studied in graph and matroid theory. Gordon was the first to study the Tutte polynomial $T(P;x,y)$ of the greedoid induced by a poset $P$, including the special case of series-parallel posets. In particular, Gordon and McMahon conjectured that, for any two series-parallel posets $P$ and $Q$, the equality $T(P;x,y)=T(Q;x,y)$ holds if and only if $P\cong Q$. In studying this conjecture, Gordon introduced a subclass $\mathcal{P}$ of series-parallel posets and proved that this equivalence holds for all $P,Q\in\mathcal{P}$. In this paper, we introduce a new subclass $\mathrm H$ of series-parallel posets and prove that $\mathcal P\subseteq\mathrm H$. Moreover, we show that, for every $P\in\mathrm H$ and every $Q\in\mathrm{SP}$, the equality $T(P;x,y)=T(Q;x,y)$ holds if and only if $P\cong Q$, thereby extending Gordon's result.

For positive integers $n,r,t$, let $\delta(n,r,t)$ denote the maximum possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and chromatic number at most $t$. Lo, Treglown and Zhao established a general upper bound for this parameter and used it, together with explicit constructions, to determine the corresponding multipartite clique threshold up to an additive constant in a broad parameter range. I determine the chromatic parameter throughout the range $r=mt-a$, $m\ge2$, $t\ge3$, $2\le a\le \min\{m,t-1\}$. The answer differs from the Lo–Treglown–Zhao upper bound by at most one. I give an explicit arithmetic criterion deciding when this one-unit correction occurs. The proof reduces the problem to an integer matrix extremum. In the boundary case, equality forces the supports of all mixed rows to form a spanning star, after which the only remaining obstruction is a divisibility condition. Combining this formula with the Andrasfai–Erdos–Sos theorem sharpens the known equality range for $f(n,r,t+1)=\delta(n,r,t)$. In particular, for $t=3$ it removes the remaining size restrictions at $r=10$ and $r=13$. Together with the $r=7$ result in arXiv:2609.19177, the classical $r=4$ case, and the known congruence classes, this gives a formula for the multipartite $K_4$-free problem for every admissible $r\ge4$ and every $n\ge1$.

We prove that the Turán density of the tetrahedron $K_4^{(3)}$ satisfies $\pi(K_4^{(3)}) \le 14993367693127837/26880000000000000 < 0.557789$, improving Baber's upper bound of $0.5615$ and closing about $62\%$ of the gap to the conjectured value $5/9$. The proof uses an exact seven-vertex flag-algebra certificate incorporating degree-stationarity from Razborov's differential method. To find the certificate, we combine the established techniques of cutting planes and column generation to optimize jointly over flag families whose types have at most five vertices. We give a complete formal proof of this Turán density bound in Lean 4.

We give snakes that are longer than the previous best known in dimensions 13 through 20 and improve the general lower bound for every dimension $d \geq 21$. Our explicit snakes reach 371,711 edges in dimension 20. Twenty compatible paths in that cube allow generalisation to give snakes of length at least $(17/48)2^d$ for every $d \geq 21$. Their controlled overlaps allow copies to be joined across the layers of a larger cube without creating shortcuts. Four additional paths give the same bound for coils. We explain the construction, prove the joining rule, and then count its length. The finite paths and their required intersections are independently verifiable.

Let $G$ be the line graph of a finite simple graph, with $n\geq1$ vertices and maximum degree $\Delta$. We prove that single-site Glauber dynamics for uniform proper $q$-colorings mixes in $O_\Delta(n\log(n/\varepsilon))$ steps for every integer $q\geq\Delta+5$. Our proof uses the Bochner framework of Chen and Liu (2026).

For a finite simple graph $G$, the matching complex $M(G)$ is the simplicial complex whose vertex set is the edge set of $G$ and whose simplices are all the matchings in $G$. The topology of the matching complex of the $m\times n$ grid graph $G_{m\times n}$ is known only for $m = 1,2$, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex $M(G_{3 \times n})$. We prove that for $n\ge2$, its reduced homology vanishes in dimensions $i \leq n-2$ and in top dimension, while $\tilde{H}_{n-1}(M(G_{3\times n}))\neq 0$. We also show that $M(G_{3 \times n})$ is simply connected for $n \geq 3$. Consequently, the topological connectivity of $M(G_{3\times n})$ is $n-2$.

We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's $3k-4$ theorem, that is every compact set $A\subseteq G$ of sufficiently small Haar measure satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained in a one dimensional Bohr set of measure at most $\mu_G(A+A)-\mu_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.

We provide an equitable partition that gives an exact, minimal representation for the graph Cartesian product formed from quantum-like bits that preserves the relevant spectral and topological properties. We show that this result follows from the fact that the operations of taking the Cartesian product of graphs and constructing equitable partition of the graphs commute. Numerical simulations illustrate the preserved emergent eigenstates in the reduced structures. Additionally, we provide a construction of the minimal structure without passing through the Cartesian product. Finally, we frame quantum-like structures in the language of topology and fibrations.

Dyer conjectured that the join of two biclosed sets of positive roots can be described by increasing Bruhat paths whose reflection labels belong to their union. We give a type-uniform proof of this conjecture for all finite Coxeter groups. More generally, for a $2$-coclosed set $C$ of positive roots contained in an inversion set, we show that its $2$-closure is an inversion set $I(w)$ and that the elements reachable from the identity using reflections labeled by roots in $C$ form exactly the set $[e,w]_B w^{-1}$. The proof combines Dyer's closure criteria with a root-selection argument.

Guo, Fang, and Lu recently proved the Komlós conjecture: for vectors $v_1,\ldots,v_n\in\mathbb R^d$ of Euclidean norm at most one, there are signs $\varepsilon_j\in\{-1,1\}$ with $\|\sum_j\varepsilon_jv_j\|_\infty\le3\sqrt{2\pi}$. We give a short proof of the bound $3\pi$ that keeps the geometric lifting framework of (Guo, Fang, and Lu 2026a) while replacing the analytic core by a quadratic Dirichlet energy. Moreover, using a more fine-grained analysis of the stability of the product-cosine function of (Smirnov and Vershynin 2026) under translations, we further sharpen the bound to below $6.9013$. On the algorithmic side, based on the polynomial-time construction of (Guo, Fang, and Lu 2026b) we give a deterministic algorithm that finds a coloring of discrepancy at most $37.54$ using at most $\widetilde O(mn+n^4)$ arithmetic operations.

A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graphs are 1-planar, we introduce the concept of $k$-flexibility: every set of at most $k$ prescribed edges can remain uncrossed in some 1-planar drawing. We use this property to reconstruct 1-planar drawings of larger graphs from drawings of smaller $k$-flexible graphs. This replaces exhaustive testing of more than forty billion cubic graphs with computations on far fewer graphs of smaller order.

Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\mathbb F_q$ in every dimension $k$ satisfying \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q, \qquad B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor. \] For fixed $p$, the first interval is linear in $q$; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic $p>0$. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at $|G|=2k-3+n$ it proves Chowdhury's full-row-rank conjecture without the $q$-dependent restriction. Specialization to $\mathbb F_q$, together with the Ball–Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power $q$, every normal rational curve in $\mathrm{PG}(N,q)$ is complete for $2\leqslant N\leqslant q-2$, and every projective Reed–Solomon code of length $q+1$ and dimension $2\leqslant k\leqslant q-2$ has covering radius $q-k$.

We study rainbow spanning configurations in bijumbled graphs whose edges are coloured independently and uniformly from a prescribed palette. For $n$-vertex $(p,\beta)$-bijumbled graphs with minimum degree at least a fixed positive multiple of $pn$, we obtain rainbow perfect matchings and Hamilton cycles with a sufficient palette surplus of order $(\log n)/p$, assuming $pn=\omega(\log n)$ and $\beta\le cpn$ for a sufficiently small constant $c$. For each prescribed spanning tree of fixed maximum degree $\Delta\ge2$, a surplus of order $L_{n,\Delta}(\log n)/p$ suffices under $\beta\le cpn/L_{n,\Delta}$, where $L_{n,\Delta}=\Delta^{5\sqrt{\log n}}$. The palette surplus is sublinear under these hypotheses. These results use a McDiarmid-type coupling and retain the discrepancy scales of the relevant deterministic embedding theorems. We also prove exact-palette results, using precisely as many colours as the number of edges of the target configuration. After independent edge percolation at rate $\rho$, it is shown that a rainbow perfect matching or Hamilton cycle exists asymptotically almost surely when $\rho pn \ge C(\log n)^2$ and $\beta\le\gamma pn$ for appropriate constants $C$ and $\gamma$. We obtain corresponding results for each prescribed bounded-degree spanning tree and for clique factors under appropriate stronger hypotheses. The exact-palette proofs construct spread measures from uncoloured containment estimates and apply the rainbow threshold theorem of Han and Yuan.