For a finite group \( G \), a bijection \( \theta: G \to G \) is a \emph{strong complete mapping} if the mappings \( g \mapsto g\theta(g) \) and \( g \mapsto g^{-1}\theta(g) \) are both bijections. A group is \emph{strongly admissible} if it admits strong complete mappings. Strong complete mappings have several combinatorial applications. There exists a Latin square orthogonal to both the multiplication table of a finite group \( G \) and its normal multiplication table if and only if \( G \) is strongly admissible. The problem of characterizing strongly admissible groups is far from settled. In this paper, we will update progress towards its resolution. In particular, we will present several infinite classes of strongly admissible dihedral and quaternion groups and determine all strongly admissible groups of order at most 31.

The complete directed graph of order \(n\), denoted \({K}_n^*\), is the directed graph on \(n\) vertices that contains the arcs \((u,v)\) and \((v,u)\) for every pair of distinct vertices \(u\) and \(v\). For a given directed graph \(D\), the set of all \(n\) for which \({K}_n^*\) admits a \(D\)-decomposition is called the spectrum of \(D\). In this paper, we find the spectrum for each bipartite subgraph of \({K}_4^*\) with 5 or fewer arcs.

A bipartite graph on \(n\) vertices, with \(n\) even, is called uniquely bi-pancyclic (UBPC) if it contains precisely one cycle of length \(2m\) for every \(2 \leq m \leq \frac{n}{2}\). In this note, using computer programs, we show that if \(32 \leq n \leq 56\), and \(n \neq 44\), then there are no UBPC graphs of order \(n\). We also present the six non-isomorphic UBPC graphs of order 44. This improves the recent results on UBPC graphs of order at most 30.

We first introduce the concept of \((k, k’, k”)\)-domination numbers in graphs, which is a generalization of many domination parameters. Then we find lower and upper bounds for this parameter, which improve many well-known results in the literature.

We are interested in ordering the elements of a subset \( A \) of the non-zero integers modulo \( n \) in such a way that all the partial sums are distinct. We conjecture that this can always be done, and we prove various partial results about this problem.

A graph \( G \) with maximum degree \( \Delta \) and edge chromatic number \( \chi'(G) > \Delta \) is \emph{edge-\(\Delta\)-critical} if \( \chi'(G-e) = \Delta \) for each \( e \in E(G) \). In this article, we provide a new proof of adjacency Lemmas on edge-critical graphs such that Vizing’s adjacency lemma becomes a corollary of our results.

This paper surveys recent results for flag enumeration of polytopes, Bruhat graphs, balanced digraphs, Whitney stratified spaces and quasi-graded posets.

A bipancyclic graph on \( v \) vertices is a bipartite graph that contains, as subgraphs, cycles of length \( n \) for every even integer \( n \) such that \( 4 \leq n \leq v \). Such a graph is uniquely bipancyclic if it contains exactly one subgraph of each permissible length.

In this paper, we find all uniquely bipancyclic graphs on 30 or fewer vertices.

A balanced complete bipartite graph is a complete bipartite graph where the degrees of its vertices differ by at most 1. In a red-blue-green coloring of the edges of a graph \( G \), every edge of \( G \) is colored red, blue, or green. For three graphs \( F_1 \), \( F_2 \), and \( F_3 \), the 2-Ramsey number \( R_2(F_1, F_2, F_3) \) of \( F_1 \), \( F_2 \), and \( F_3 \), if it exists, is the smallest order of a balanced complete bipartite graph \( G \) such that every red-blue-green coloring of the edges of \( G \) contains a red \( F_1 \), a blue \( F_2 \), or a green \( F_3 \). In this note, we determine that

\[
20 \leq R_2(C_4, C_4, C_4) \leq 21.
\]

A Hamiltonian graph \( G \) is said to be \(\ell\)-path-Hamiltonian, where \(\ell\) is a positive integer less than or equal to the order of \( G \), if every path of order \(\ell\) in \( G \) is a subpath of some Hamiltonian cycle in \( G \). The Hamiltonian cycle extension number of \( G \) is the maximum positive integer \(\ell\) for which every path of order \(\ell\) or less is a subpath of some Hamiltonian cycle in \( G \). If the order of \( G \) equals \( n \), then it is known that \( \text{hce}(G) = n \) if and only if \( G \) is a cycle or a regular complete bipartite graph (when \( n \) is even) or a complete graph. We present a complete characterization of Hamiltonian graphs of order \( n \) that are \(\ell\)-path-Hamiltonian for each \(\ell \in \{n-3, n-2, n-1, n\}\).