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.

Clustering a signed graph means partitioning the vertices into clusters so that every positive edge, and no negative edge, is within a cluster. The obstruction to clustering is circles with exactly one negative edge (“weakly negative circles’’). The correlation clustering problem is to cluster with the minimum number \(Q\) of edges that violate the clustering rule. A lower bound is \(w\), the maximum number of edge-disjoint weakly negative circles. If every two such circles are edge disjoint, then \(Q=w\). We characterize the signed graphs in which no two weakly negative circles share any edges. A corollary is a straightforward recognition algorithm for such signed graphs. An unsolved problem is to characterize the signed graphs with \(Q=w\).

Recently, Luo [3] introduced the Adding–Swapping Mapping Method to provide an alternative and constructive proof of Stanley’s [4] conjecture on perfect square permutations in \(S_n\) and asked whether the method extends to higher powers. In this paper we answer that question in a more limited but precise structural sense. For each fixed \(k\ge 2\), we define the \(k\)-signature \(R_k(w)\) recording the cycle-count vector modulo \(\gcd(m,k)\) in each length \(m\), and we prove a local residue transition law describing how the insertion map \(D_i\) updates the signature once the cycle length of the insertion point is specified. We also prove explicitly that every \(k\)-th power permutation has zero \(k\)-signature, so the signature gives a necessary obstruction to being a \(k\)-th power. This yields a residue-based partition of \(S_n\) that serves as an indexing scheme for insertion updates. We then show that for \(k\ge 3\) the insertion family does not preserve the class of \(k\)-th powers, explaining why the square case is exceptional from the standpoint of Luo’s method. Finally, we include explicit small-\(n\) data for \(k=3,4\) and prove that the density of \(k\)-th powers in \(S_n\) tends to \(0\) as \(n\to\infty\).

This article studies edge irregular \((k)\)-labelings of complete graphs \((K_n)\) and aims to improve the existing upper bound for the edge irregularity strength \((es(K_n))\) through an algorithmic approach. The improvement is achieved using the branch and bound algorithm design strategy, whose selection is important because of the problem structure, computational complexity, possible serial or parallel execution, and accuracy requirements. Labeling complete graphs is difficult because the number of edges grows rapidly and many triangles are involved, making it challenging to maintain unique edge weights while searching for optimal labels. Since complete graphs serve as supergraphs for many graph families, results on them are particularly valuable. In 2018, Asim et al. proposed an algorithmic solution for complete graphs and gave the upper bound \((es(K_n)\leq E\log_2 |V|)\). This article uses the branch and bound strategy to address edge deficiency and improve that bound. Computational experiments are conducted for higher-order graphs, where the algorithm recursively generates constrained combinatorial structures to ensure unique edge weights. The results show that the proposed branch and bound algorithm significantly improves the upper bound for \((es(K_n))\) compared with previous results.