The triplication method for constructing strong starters in \(\mathbb{Z}_{3m}\) from starters in \(\mathbb{Z}_{m}\) (say, a starter of order 21 from a starter of order 7) was proposed by the authors in 2025. The method reduced the construction of this particular combinatorial design (a strong starter in a cyclic group) to solving a Sudoku-type problem – an independent task with its own tools and techniques available. The Sudoku-type problem was formulated in terms of the so-called triplication table constructed from a starter of order \(m\). The method was applicable to odd orders \(m\ge 7\) not divisible by 3. In the present paper, our previous approach is developed in two directions: (1) the definition of the triplication table is generalized, which expands possibilities for its construction to include three base starters, “pseudostarters”, or even more general setup; (2) the formulation of the Sudoku-type problem is broadened to embrace various scenarios of “modular encoding” and reconstruction of strong starters from its solution. A theoretical gain of these developments is an improved understanding of the general structure of the triplication approach. A practical outcome is that all odd values \(m \ge 5\) (including those divisible by 3) are now admissible and the set of possible triplication tables is so broad that any latent strong starter of odd order \(3m\) can emerge by triplication.

For the prism graphs \(G_n=C_n\square P_2\), the chromatic polynomial has an explicit four-branch transfer-matrix expansion with polynomial eigenvalues and amplitudes. The Beraha-Kahane-Weiss (BKW) theorem then confines asymptotic root accumulation to equimodular ties and amplitude zeros. Here the only amplitude zeros are \(z=1\) and \(z=\frac{3\pm\sqrt5}{2}\), and a dominance check shows that none yields an isolated BKW limit point. A complete algebraic classification of the prism tie curves appears in [1]. The dominant quadratic-linear ties are recast here in a centered Cassini-type normal form, giving a product-of-distances interpretation together with explicit quartic implicit and centered polar equations. A global Rouché comparison also yields a uniform finite-\(n\) bound: every chromatic root of \(G_n\) satisfies \(|z|<6\) for all \(n\ge3\).

For a fixed integer \(k\geq0\), let \(p_k(n)\) denote the number of integer partitions \(\lambda=(\lambda_1,\ldots,\lambda_r)\) of \(n\) satisfying the minimal-difference condition \(\lambda_i-\lambda_{i+1}\geq k,\quad 1\leq i<r,\) with the convention that the smallest part is at least one. We study the logarithmic asymptotic growth of \(p_k(n)\) through the length-refined generating function \(P_k(q)=\sum_{r\geq0}\frac{q^{\,r+k\binom r2}}{(q;q)_r}.\) The factor \(q^r\) is essential and comes from the condition that each part is positive. For \(k\geq1\), we prove that \(\log p_k(n)\sim B_k\sqrt n,\) where \(B_k=2\sqrt{A_k}\) and \(A_k=\frac{\pi^2}{6}-Li_2(e^{-x_k})-\frac{k}{2}x_k^2,\) with \(x_k>0\) the unique solution of \(1-e^{-x_k}=e^{-kx_k}\). The case \(k=0\) is stated separately and gives the classical Hardy–Ramanujan constant \(B_0=\pi\sqrt{2/3}\). The proof combines a uniform Euler–Maclaurin estimate for the truncated Euler product, a discrete Laplace principle for the length sum, and Ingham’s Tauberian theorem.

The enumeration of noncrossing trees has attracted significant attention since the turn of 21st century. These trees have been studied with respect to various statistics, including the number of vertices, leaves, root degree, and levels. In contrast, plane trees have a longer history of exploration. In 2010, Deutsch and his co-authors introduced and enumerated a class of plane trees in which a rightmost edge may be marked, provided it does not lead to a leaf. Their enumeration formula involved the Catalan numbers, which also count plane trees. In this work, we extend the concept of marking rightmost edges to noncrossing trees, introducing a new combinatorial structure. We enumerate this structure according to the number of edges, marked edges, root degree, and leaves. We use symbolic method and Lagrange Inversion Formula to derive our results. Furthermore, we establish connections between these new structures and both labelled plane trees and ternary trees.

Recently, the notion of a Weyl substructure in a spherical building was introduced type by type. In this paper we provide a uniform (axiomatic) definition across all types. In particular, this provides a new characterisation of the Ree-Tits octagons. We then show that uniclass automorphisms of spherical buildings are uniformly characterised by their fix structure. For type preserving automorphisms, this follows from earlier work, and so the focus here is on dualities. In particular, it follows that a duality pointwise fixing a Weyl substructure is automatically a polarity. This characterises all polarities in self-dual spherical buildings where opposition acts trivially on the types.

We introduce the notion of a move graph, that is, a directed graph whose vertex set is a \(\mathbb Z\)-module \(\mathbb Z_n^m\), and whose arc set is uniquely determined by the action \(M\!:\!\mathbb Z_n^m\to \mathbb Z_n^m\) where \(M\) is an \(m\times m\) matrix with integer entries. We study the manner in which properties of move graphs differ when one varies the choice of cyclic group \(\mathbb Z_n\). Our principal focus is on a special family of such graphs, which we refer to as “sub-add move graphs.”

Given functions \(f,g: [n] \rightarrow [n]\), do there exist \(n\) points \(A_1,A_2,\ldots,A_n\) in some metric space such that \(A_{f(i)},A_{g(i)}\) are the points closest and farthest from point \(A_i\)? In this paper we characterize precisely which pairs of functions have this property. Define \(m(k)\) to be the maximum integer such that any pair of functions \(f,g:[m(k)]\rightarrow [m(k)]\) realizable in some metric space is also realizable in \(\mathbb{R}^k\). We show that \(m(k)\) grows exponentially in \(k\). This answers a question of Croft. We also discuss what happens when looking at minimum and maximum distances separately.

Suppose \(G\) is a graph and \(L\) is a list assignment for \(G\). A request of \(L\) is a function \(r\) with nonempty domain \(D\subseteq V(G)\) such that \(r(v) \in L(v)\) for each \(v \in D\). The triple \((G,L,r)\) is \(\epsilon\)-satisfiable if there exists a proper \(L\)-coloring \(f\) of \(G\) such that \(f(v) = r(v)\) for at least \(\epsilon|D|\) vertices in \(D\). We say \(G\) is \((k, \epsilon)\)-flexible if \((G,L’,r’)\) is \(\epsilon\)-satisfiable whenever \(L’\) is a \(k\)-assignment for \(G\) and \(r’\) is a request of \(L’\). It is known that a graph \(G\) is not \((k, \epsilon)\)-flexible for any \(k\) if and only if \(\epsilon > 1/ \rho(G)\) where \(\rho(G)\) is the Hall ratio of \(G\). The list flexibility number of a graph \(G\), denoted \(\chi_{\ell flex}(G)\), is the smallest \(k\) such that \(G\) is \((k,1/ \rho(G))\)-flexible. A fundamental open question on list flexibility numbers asks: Is there a graph with list flexibility number greater than its coloring number? In this paper, we show that the list flexibility number of any complete multipartite graph \(G\) is at most the coloring number of \(G\). We also initiate the study of list epsilon flexibility functions of complete bipartite graphs which was first suggested by Kaul, Mathew, Mudrock, and Pelsmajer in 2024. Specifically, we completely determine the list epsilon flexibility function of \(K_{m,n}\) when \(m \in \{1,2\}\) and establish some additional bounds for small \(m\). Our proofs reveal a connection to list coloring complete bipartite graphs with asymmetric list sizes which is a topic that was explored by Alon, Cambie, and Kang in 2021.

A \(k\)-edge coloring \(c\) of the edge set \(E (G)\) of a graph \(G\) is a surjective mapping \(c : E (G) \to [k] = \{1, 2, \ldots, k\}\). If \(\mathcal{F}\) and \(\mathcal{H}\) are families of graphs, \(MRS(K_n; \mathcal{F}, \mathcal{H})\) is the set of numbers \(k\) such that there is a \(k\)-edge coloring of \(K_n\) with respect to which there is neither a monochromatic copy of any \(F \in \mathcal{F}\) nor a rainbow copy of any \(H \in \mathcal{H}\) in \(K_n\). Our main result is that for all \(n \geq 2\), \(MRS(K_n;\{\text{odd cycles}\},\{\text{cycles}\}) = \{\lceil \log_2 n \rceil, \ldots, n – 1\}\). The proof will exploit an idea for edge-coloring connected graphs so as to forbid rainbow cycles to be found in [4].

If \(S\) is a numerical semigroup, we will denote by \({\mathrm F}(S),\) \({\mathrm g}(S)\) and \({\mathrm t}(S),\) the Frobenius number, the genus and the type of \(S,\) respectively. We will also denote by \({\mathrm n}(S)\) and \({\mathrm i}(S)\) the cardinality of the sets \(\{s\in S\mid s<{\mathrm F}(S)\}\) and \(\{x\in \mathbb{N}\backslash S\mid x-1\in S\},\) respectively. In this paper we will study the \(\mathrm{PTT}\)-semigroups. That is, perfect numerical semigroups with type two. In particular, we will see that if \(S\) is a numerical semigroup, then the following conditions are equivalent: 1) \(S\) is a \(\mathrm{PTT}\)-semigroup; 2) The set of pseudo-Frobenius numbers of \(S\) is \(\{{\mathrm F}(S),{\mathrm F}(S)-1\}\); 3) \(S\) is maximal in the set \(\{T\mid T \mbox{ is a numerical semigroup } T\cap \{{\mathrm F}(S),{\mathrm F}(S)-1\}=\emptyset \mbox{ and } {\mathrm t}(T)=2\}\); and 4) \({\mathrm F}(S)-1\notin S\) and \({\mathrm n}(S)={\mathrm g}(S)-{\mathrm i}(S).\) As an application of these characterizations, we will provide several algorithms for calculating all the \(\mathrm{PTT}\)-semigroups with a given Frobenius number.