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 number 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.

In finite desarguesian projective planes of prime power order \(q\), we consider the MDS codes obtained from ovals and generate new codes for the purpose of permutation decoding. We show that those new codes are \([q^2-1,3,(q-1)^2]_q\) codes and have \(s\)-PD-sets for \(s\le q-1\).

In this paper, we explore the enumerative combinatorics of American-style crossword puzzle grids under modified answer length requirements. While standard American-style crossword rules have a minimum answer length of three cells, we generalize this constraint to a minimum length \(m\) for an \(n\times n\) grid. We define \(\lvert Puz (n,m)\rvert\) as the number of such grids satisfying the standard structural rules of connectivity, \(180^\circ\) rotational symmetry, keyed squares, and full dimensionality. We prove that for \(m>\frac{n}{2}\), the number of valid grids is invariant under the transformation \(\lvert Puz (n,m)\rvert=\lvert Puz (n+1,m+1)\rvert\). Furthermore, we establish a closed-form formula for \(\lvert Puz (n,m)\rvert\) when \(m>\frac{n}{2}\). We also verify some counts for smaller grid dimensions verifying previously conjectured values.

Let \(G\) be a graph of order \(n\), maximum degree at most \(\Delta\), and no component of order 2. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of \(G\) as a function \(\rho : V(G) \to \mathbb{N}_{0}\) for which \[\sigma : V(G) \to \mathbb{N}_{0} : u \mapsto (1+\rho(u)) d_G(u) + \sum\limits_{v\in N_G(u)} \rho(v),\] is a vertex coloring, that is, adjacent vertices receive different values under \(\sigma\). They show the existence of a proper pushing scheme \(\rho\) with \(\max\{\rho(u) : u \in V(G)\} \leq \Delta^2\) and conjecture that this upper bound can be improved to \(\Delta\). We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme \(\rho\) with \(\sum\limits_{u\in V(G)} \rho(u) \leq (2\Delta^2+\Delta)n/6\).

A strongly connected digraph \(D\) is primitive provided the greatest common divisor of the lengths of its directed cycles equals 1. The scrambling index of a primitive digraph \(D\) is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex \(w\) such that we can get to \(w\) from \(u\) and \(v\) in \(D\) by directed walks of length \(k\). In this paper, we characterize those primitive doubly symmetric digraphs with the largest scrambling index.