Papers

We define almost affine vector rank-metric codes as subsets $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ whose canonical projections have cardinalities that are powers of $q^m$, and prove that they naturally induce $q$-matroids. We establish that the operations of puncturing and shortening correspond to restriction and contraction of the $q$-matroid, and show that the rank-weight and formal dual distance distributions are determined by the induced $q$-matroid. We briefly discuss applications to perfect $q$-matroid ports in linear network coding, and show that disconnected $q$-matroids need not induce disconnected ports. Finally, we show that certain Additive Generalized Twisted Gabidulin codes yield direct examples of strictly almost affine rank-metric codes, alongside a separate construction derived from proper finite semifields.

We prove Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs, where the co-claw is the disjoint union of a triangle and a vertex, the co-gem is the disjoint union of a 4-vertex path and a vertex, the fork is obtained from $K_{1,3}$ by subdividing one of the edges, and the antifork is the complement of the fork. The $\{\text{co-claw}, \text{co-gem}\}$-free graphs include the complements of line graphs of triangle-free multigraphs, and thus our results imply Hadwiger's Conjecture for these graphs. In fact, we prove a stronger result: every $\{\text{co-claw}, \text{co-gem}\}$-free graph $G$ has a $K_{\chi(G)}$-model where each branch set has size at most 2, and every $\{\text{fork}, \text{antifork}\}$-free graph $G$ has a $K_{\chi(G)}$-model where at most one branch set has size greater than 2.

A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the property of being a defining set is retained. Equivalently, a $k$-strong defining set intersects every Latin trade in the Latin square at least $k$ times. In the addition table for integers modulo $n$, when $n$ is even we determine the minimum size of a $k$-strong defining set for any $k$. For odd $n$ we give a construction for a minimally $2$-strong defining set. We furthermore give computational results for Latin squares of small orders.

We prove two positivity conjectures proposed by Guo for alternating sums and factorial ratios built from Gaussian coefficients. The first result proves the positivity of the odd $q$-super Catalan numbers \[ C_{m,n}(q)=\frac{[2m+1]![2n]!}{[m+n+1]![m]![n]!}. \] The proof uses the positivity theorem of Warnaar and Zudilin for the usual $q$-super Catalan numbers, together with two recurrences obtained from a double application of the $q$-Chu–Vandermonde summation. The second result proves Guo's conjectural strengthening of his alternating-sum positivity theorem, replacing the exponent coefficient $2r-1$ by every odd coefficient $2b-1$, $1\leq b\leq r$. Its proof combines a $q\mapsto q^{-1}$ reciprocity with a finite deletion recurrence.

For an integer $\ell\geq 2$, let ${\cal{H}}_{\ell}$ denote the family of graphs which have girth $2\ell$ and have no even hole of length greater than $2\ell$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{\ell\geq 2} {\cal{H}}_{\ell}$ is $3$-colorable. Chen showed that every graph in $\bigcup_{\ell\geq 5} {\cal{H}}_{\ell}$ is $3$-colorable. In this paper, we prove that every graph in ${\cal{H}}_4$ is $3$-colorable.

The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ such that $\|s_i t\|\geq \frac{1}{k+1}$ for all $i$. We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\mathbf d=(d_1,\ldots,d_k)\in(0,1/2]^k$, let $\mathsf{MLPS}_k$ be the set of vectors such that, for every choice of distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ with $\|s_i t\|\geq d_i$ for all $i$. We give an exact characterization of $\mathsf{MLPS}_2$. We also use Fourier series for distance-threshold indicator functions to obtain an arithmetic progression summation formula and an exact two-function integral formula for unequal thresholds.

A sprout sequence is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq 0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is called the seed of $\frakr$. This concept originated in the work of Littlewood and Richardson (though not with the name “sprout sequence”), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case $F(t)=\sec(\sqrt{t})$. We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of $R_n$ for all $n$ are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The expansion of $R_n$ in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for $R_n$ as a sum of chromatic symmetric functions of interval orders.

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded pathwidth and bounded maximum degree has a tree-partition of bounded width, with the extra property that the underlying tree has bounded pathwidth. Moreover, we prove a lower bound showing that the bound on the pathwidth of the underlying tree is within a constant factor of optimal.

The Mycielskian is a standard construction studied in many an introductory graph theory course. It is natural to consider Mycielskians of cycles, some of the simplest of all graphs. This paper deals with the so-called “dimension” of such graphs. The dimension of a graph $G$ is the smallest positive integer $n$ such that there exists a one-to-one correspondence between the vertices of $G$ and some collection of points in $n$-dimensional Euclidean space such that if two vertices in $G$ are adjacent, then the distance between the corresponding points is $1$. In previous works, it had been proven that the dimension of the Mycielskian of a $k$-cycle is 3 when $k$ is $3$, $4$, or $5$, and 2 when $k=10$. In this paper, we answer the question completely. Namely, we show that the dimension is $3$ when $k\neq 10$, and $2$ when $k=10$.

Block designs are combinatorial structures in which each pair of a set of varieties appears together in a fixed number of blocks. Complete graphs are graphs in which every pair of vertices are adjacent. We present some new constructions of block designs using complete graphs, including two infinite families of designs using edge sets of complete graphs.

We give a complete classification of the Jordan types occurring in the nilpotent commutator of a nilpotent matrix whose Jordan type is a hook partition. As a consequence, we also show that two partitions with the same generic commuting Jordan type need not commute with each other.

We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $\eta'_p(G)$, is equal to its chromatic index. This verifies that a strengthening of the Additive Coloring Conjecture of Czerwiński et al. holds for line graphs of $d$-regular Vizing-class-1 graphs. We show that if $G$ is a $d$-regular Vizing-class-2 graph, $\eta'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, and if $G$ is a $d$-regular Vizing-class-2 graph that admits a proper edge-coloring with a smallest color class of size $r$ and $\text{girth}(G)\geq 6r-5$, then $\eta_p'(G)\leq 2d$, among other results.

The picture-hanging puzzle, popularized by Demaine et al. (2014), asks for a way to wrap a wire around $n$ nails such that the picture hangs as long as fewer than $k$ nails are removed, but falls as soon as any $k$ are removed. Solutions correspond to words in the free group $F_n$. We give explicit, deterministic, polynomial-length constructions for two regimes: $2$-out-of-$n$ with word length at most $\tfrac{8}{3}n^{\log_2 6} - 4n^2$, and $(n-2)$-out-of-$n$ with word length $6n\log_2(n/2)$, both for $n$ a power of two. These improve on Wästlund's quasi-polynomial deterministic construction in their respective regimes. We also report, via exhaustive computer search, the exact minimum length of $16$ for the $2$-out-of-$4$ puzzle, attained by two structurally distinct solutions. As an additional contribution, we observe that the natural workshop realization with carabiners on a flat board introduces an over/under ambiguity at every wire crossing; a wrong choice can produce a Whitehead link, which is topologically distinct from the intended commutator.

We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jordan curve has inflection minimum zero. For tree-like shadows, inflection criterion converts inflection-free realizability into a finite coorientation problem on the building polygons of the shadow. We sharpen this viewpoint into an exact finite formula for the minimum number of normalized inflections and record a dynamic-programming computation on the block tree. We then push the method beyond the tree-like case. For every embedded shadow the same coorientation model gives a universal lower bound. For a natural larger class, called tree–necklace shadows, in which the non-tree-like blocks are separated annular cycles, the lower bound is exact after imposing an explicit $\mathbb Z_2$ holonomy condition around each necklace. We also record the algorithmic status of the exact minimization problem and formulate a likely NP-hardness problem for unrestricted shadows. Finally, we introduce a related invariant: the minimum possible least multiplicity of the Gauss map, equivalently the smallest guaranteed number of oriented parallel tangencies. This “parallel-tangent load” is controlled by the same inflection folds but is not determined by their number alone.

Let $\mathcal{T}$ be a finite nonempty set of $3$-element subsets of a totally ordered set $V$. We view $\mathcal{T}$ as the set of triangles in the support graph. Let $\delta_{1,\mathcal{T}}$ be the signed edge-triangle incidence matrix, and $\lambda(\mathcal{T})$ the spectral gap of $\delta_{1,\mathcal{T}}^T\delta_{1,\mathcal{T}}.$ Our main results show that large $\lambda(\mathcal{T})$ forces strong overlap and a large minimum degree in the support graph. In particular, every support edge lies in at least $\lceil \lambda(\mathcal{T})\rceil-2$ triangles in $\mathcal{T}$ and hence the graph has minimum degree at least $\lceil \lambda(\mathcal{T})\rceil-1$. We further prove that $\binom{n}{3}$ is the exact threshold for attaining level $n:$ if $|\mathcal{T}|< \binom{n}{3}$, then $\lambda(\mathcal{T}) \leq n-1,$ while if $|\mathcal{T}|=\binom{n}{3}$ and $\lambda(\mathcal{T}) > n-1,$ then $\mathcal{T}$ is exactly the full set of triangles on an $n$-vertex clique. Moreover, this clique peak is isolated in a strong interval-scale sense: letting $\phi(t)=\max_{|\mathcal{T}|=t} \lambda(\mathcal{T})$, immediately above $\binom{n}{3}$ there is a forbidden interval on which $\phi(t) \leq n-1$, and the first passage above the level $n-1$ is delayed by $\Theta(n^2)$ additional triangles. Since $\binom{n+1}{3} - \binom{n}{3}=\Theta(n^2),$ this implies that after the peak at $\binom{n}{3}$ one must traverse a nonzero proportion of the full gap until the next clique threshold before substantial recovery can occur. In particular, $\phi$ is not monotone. However, $\phi(t)=\Theta(t^{\frac{1}{3}}).$ Finally, if $\Lambda(t):=\max_{1 \leq s \leq t}\phi(s),$ then $\Lambda(t)=\max\{n \in \mathbb{N}:\binom{n}{3} \leq t\}.$ Thus complete triple systems are the unique minimal spectral extremizers, but their peaks are isolated on the natural scale between consecutive clique thresholds.

Lie algebra $\mathfrak{sl}(2)$ can be realised by vector fields on $\mathbb{R}^1\ni x$ with polynomial coefficients $1$, $-2x$, $-x^2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x^k/k!$, $\ldots$, $x^N/N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $\mathbf{1} \wedge \partial_x \wedge \ldots \wedge \partial_x^{N-1}$ as the $N$-ary brackets. Over dimension $d=2$ with $\mathbb{R}^2\ni(x,y)$ and for the complete generalised Wronskian $W_{d=2}^{k=1}=\mathbf{1}\wedge \partial_x \wedge \partial_y$ of differential order $k=1$ as the ternary bracket, the finite-dimensional polynomial SH-Lie algebras are spanned by $\langle 1$, $x$, $y$, $p\rangle$ with $p\in\{x^2$, $xy$, $y^2\}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras $\Bbbk_k[{\boldsymbol{x}}]\subseteq \mathcal{A} \subseteq \Bbbk[x^1,\ldots,x^d]$ (over $\Bbbk=\mathbb{R}$ or $\mathbb{C}$) with the complete generalised Wronskians $W_{d\geqslant 1}^{k\geqslant 1}$ of order $k$ as $N$-ary brackets: $N=\binom{d+k}{d}$. We obtain a factorisation formula for the generalised Vandermonde determinants which show up in the structure constants of the polynomial algebras $\mathcal{A}$.

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting object, the $(d+2)$-angulated cluster category, and the $d$-almost positive subcategory of the derived category (the higher analogue of the category of two-term complexes of projectives). We show that the third one, introduced by the second-named author, is the $d$-exangulated quotient of the other two, introduced by Oppermann and Thomas, by the ideals generated by morphisms factoring through morphisms from injective to projective objects, thus providing an algebraic connection between the two models of Oppermann-Thomas. This is a $d$-exangulated version in type $A$ of a result of Brüstle and Yang and its interpretation by the first-named author together with Fang, Palu, Plamondon and Pressland. It also explains a well-known coincidence between the number of 2-term silting complexes in type $A_{n}$ and of tilting modules in type $A_{n+1}$ from the $0$-Auslander perspective. We expect this to serve as a prototypical example of the $0$-Auslander correspondence in higher homological algebra.

We prove the finite abelian two-generator conjecture of Darijani–Miraftab–Witte Morris: every directed Cayley digraph on a finite abelian group with two distinct nonzero generators has two arc-disjoint Hamiltonian paths. The proof uses a cut-reflection theorem for Hamiltonian cut values in the family Cay(Z_k; a, a+1): if Z is the set of such values and N=k-1, then, with N-Z={N-z : z in Z}, dist(Z,N-Z)<=1. The proof uses sector-filling inequalities for primitive-ray multiplicities and an extremal graph recording pairs at minimal reflected distance. The estimate is sharp modulo parity: exact reflection occurs for odd k, while distance one occurs for even k. The second remaining cyclic family, Cay(Z_k; -a, a+1), is treated by an explicit quotient–fiber construction. We also prove the remaining three-factor case for Cartesian products of directed cycles. Together with the two-factor and at-least-four-factor theorems of Darijani–Miraftab–Witte Morris, this resolves their directed-cycle product conjecture for all numbers of factors.

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon} N^{-k/d+\epsilon}. \] The proof combines Schmidt's Subspace Theorem with an explicit inhomogeneous transference argument. This improves Iyer's (2025) higher-root exponent $(k-d+1)/d^2$, and also the analogous $d$-ary full-basis exponent away from the cases where $k+1$ is a power of $d$, at the cost of ineffectivity. We also record a conjectural uniform exponent $k-1/d$. In the square-root case $d=2$, we give explicit integer-target constructions for $k=2,3,4$ attaining this conjectural value.

Problem 8.1 in Astaiza et. al. asks about the relationship between the cycle decomposition of a permutation $\sigma$ and that of its symmetric tensor power $\sigma ^{\odot k}$. In this paper, we investigate this question and give formulas for computing the number of fixed points and, in the case of a permutation containing at most one cycle of length greater than one, the number of $s$-cycles.

This paper investigates the new notion of $2$-word-$\pi$-repre\-sentable graphs: the nodes of the graph correspond to the letters of the two words and there exists an edge between two nodes if the projections of any two letters of both words are equal. The benefit of not only using one word for a representation as introduced by Kitaev and Pyatkin is that every graph is $2$-word-$\pi$-representable. We present an algorithm that returns two representing words for any graph. Aside, we show that every permutation graph is representable by two $1$-uniform words and give constructions how graph operations on $2$-word-$\pi$-representable graphs can be realised on their representing words which give further insights into the representation of cographs.

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with minimum degree to be $k$-factor-critical. A graph $G$ is fractional $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a fractional perfect matching. Clearly, the fractional $k$-factor-criticality of a graph is a necessary property for a graph to be $k$-factor-critical. Jia, Fan and Liu [Discrete Appl. Math. 386 (2026) 255-263] proposed a tight sufficient condition in terms of the spectral radius for a graph with fractional $k$-factor-criticality to be $k$-factor-critical. A natural question arises: can we derive analogous sufficient conditions by incorporating the minimum degree parameter of graphs? We first establish a lower bound on the size to ensure that a $(k+1)$-connected graph with fractional $k$-factor-criticality is $k$-factor-critical, where $k$ is a positive integer with $k\geq1$. Moreover, we provide a sufficient condition in terms of the spectral radius for a $(k+1)$-connected graph with fractional $k$-factor-criticality to be $k$-factor-critical. Our results generalize the result of Jia, Fan and Liu to $(k+1)$-connected graphs. Furthermore, our spectral conditions apply to a broader family of connected graphs compared with the results of Fan and Lin, as well as Jia et al.

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $\chi_{r}(G)=\min\{r,\Delta(G)\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $\chi_{L,r}(C_{n})=\chi_{r}(C_{n})$. The main results are the following: $(1)$ If $G$ is a tree, then $\chi_{L,r}(G)=\min\{r,\Delta(G)\}+1$; $(2)$ Let $G$ be a unicyclic graph which is not isomorphic to the cycle $C_{n}$. If $n\neq 5$ and $r\geq3$, then $\chi_{L,r}(G)=\min\{r,\Delta(G)\}+1$; otherwise, $\min\{r,\Delta(G)\}+1\leq\chi_{L,r}(G)\leq\min\{r,\Delta(G)\}+2$.

We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erdős-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{\Theta(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eliáš and Matoušek and the monotone Ramsey numbers of Balko.

Given an acyclic directed graph (DAG), the space of strength one flows is a lattice polytope called the flow polytope of the DAG. If the DAG admits an ample framing, then the flow polytope is Gorenstein and it linearly projects onto a reflexive polytope called the $\mathbf{g}$-polytope. We provide a combinatorial characterization of amply framed DAGs that have a locally anti-blocking $\mathbf{g}$-polytope, and we characterize the minimal faces of the $\mathbf{g}$-polytope containing a fixed pair of vertices. We prove in this case that the unimodular triangulation of the $\mathbf{g}$-polytope induced by the DKK triangulation of the flow polytope is a pulling triangulation, and we characterize the pulling orders that yield the DKK triangulation. To prove our results, we introduce and study coherence diagrams, a combinatorial model of coherence for amply framed DAGs with locally anti-blocking $\mathbf{g}$-polytopes. We conclude by indicating possible extensions of these results to the setting of $\mathbf{g}$-polytopes for gentle Nakayama algebras.

We construct a word-theoretic framework for generalized Markov numbers, that is, positive integers appearing in positive integer solutions of the generalized Markov equation $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$. For each positive rational slope $t$, we define a word $\omega_t$ by a recursive rule on a binary tree and realize it geometrically by a line segment of slope $t$. Matrix evaluation of $\omega_t$ gives a Markov–monodromy matrix encoding the generalized Markov number at $t$. We also show that $\omega_t$ recovers the classical Cohn word by a local substitution rule, and that the completed word $\overline{\omega}_t=xyz\omega_t^{-1}$ is related to the generalized Cohn matrices.

Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author. Preorder polytopes are shown to be lattice polytopes which satisfy a certain duality relating their Ehrhart polynomials with the zeta polynomials of their posets of lattice points. A combinatorial interpretation of the normalized volume of a preorder polytope is proven, together with formulas for the Ehrhart polynomial and the $h^\ast$-polynomial, and a combinatorial interpretation of the latter is conjectured. Several conjectures and results on the lattice point enumeration of arbor polytopes are generalized to preorder polytopes, new conjectures are proposed and new interesting examples of preorder polytopes are studied.

We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_1t+\cdots+h_st^s$. We prove that if $\Bbbk[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$, then $\Bbbk[G]$ is Gorenstein. Equivalently, under these assumptions the $h$-polynomial of $\Bbbk[G]$ is palindromic. The proof treats the $2$-connected case first by translating the numerical condition $h_1=h_{s-1}$ into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise minimal Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the $h$-polynomial.

We study signed generalized Stirling polynomials $P_k(m,x)$ arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma–polygamma formulas for $P_{m-s}(m,x)$, a central vanishing identity, a finite approximation to $\cosh \pi x$, and a limit formula for $\pi$. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner–Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence $\chi_n$. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for $\chi_j$ yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.

For graph $G$, a connected graph $H$ of order $n$ is $G$-good if $r(G,H)=(\chi(G)-1)(n-1)+s(G)$, where $\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\chi(G)$-coloring of $G$. Let $K_{\alpha_{1},\ldots ,\alpha_{p},n}$ be the complete $(p+1)$-partite graph with partite sets of sizes $\alpha_1,\ldots,\alpha_p,n$. Burr, Faudree, Rousseau and Schelp (1983) showed that $K_{\alpha_1,\ldots,\alpha_p,n}$ are $(K_2+mK_1)$-good for large $n$. We determine graphs $G$ such that $K_{\alpha_{1},\ldots ,\alpha_{p},n}$ are $G$-good for large $n$. The characterization depends on $\mathrm{snd}(\alpha_i)$, the smallest non-divisor of $\alpha_i$, where $1\le i\le p$.