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 \(\ell \in \{n-3, n-2, n-1, n\}\).

Let \( G \) be an edge-colored connected graph. A path \( P \) is a proper path in \( G \) if no two adjacent edges of \( P \) are colored the same. An edge coloring is a proper-path coloring of \( G \) if every pair \( u, v \) of distinct vertices of \( G \) is connected by a proper \( u-v \) path in \( G \). The minimum number of colors required for a proper-path coloring of \( G \) is the proper connection number \( \text{pc}(G) \) of \( G \). We study proper-path colorings in those graphs obtained by some well-known graph operations, namely line graphs, powers of graphs, coronas of graphs, and vertex or edge deletions. Proper connection numbers are determined for all iterated line graphs and powers of a given connected graph. For a connected graph \( G \), sharp lower and upper bounds are established for the proper connection number of (i) the \( k \)-iterated corona of \( G \) in terms of \( \text{pc}(G) \) and \( k \), and (ii) the vertex or edge deletion graphs \( G-v \) and \( G-e \), where \( v \) is a non-cut-vertex of \( G \) and \( e \) is a non-bridge of \( G \), in terms of \( \text{pc}(G) \) and the degree of \( v \). Other results and open questions are also presented.

A red-blue coloring of a graph \( G \) is an edge coloring of \( G \) in which every edge of \( G \) is colored red or blue. Let \( F \) be a connected graph of size 2 or more with a red-blue coloring, at least one edge of each color, where some blue edge of \( F \) is designated as the root of \( F \). Such an edge-colored graph \( F \) is called a color frame. An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \). The \( F \)-chromatic index \( \chi_F'(G) \) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). A minimal \( F \)-coloring of \( G \) is an \( F \)-coloring with the property that if any red edge of \( G \) is re-colored blue, then the resulting red-blue coloring of \( G \) is not an \( F \)-coloring of \( G \). The maximum number of red edges in a minimal \( F \)-coloring of \( G \) is the upper \( F \)-chromatic index \( \chi_F”(G) \) of \( G \). For integers \( k \) and \( m \) with \( 1 \leq k < m \) and \( m \geq 3 \), let \( S_{k,m} \) be the color frame of the star \( K_{1,m} \) of size \( m \) such that \( S_{k,m} \) has exactly \( k \) red edges and \( m-k \) blue edges. For a positive integer \( k \), a set \( X \) of edges of a graph \( G \) is a \( \Delta_k \)-set if \( \Delta(G[X]) = k \), where \( G[X] \) is the subgraph of \( G \) induced by \( X \). The maximum size of a \( \Delta_k \)-set in \( G \) is referred to as the \( k \)-matching number of \( G \) and is denoted by \( a_k'(G) \). A \( \Delta_k \)-set \( X \) is maximal if \( X \cup \{e\} \) is not a \( \Delta_k \)-set for every \( e \in E(G) – X \). The minimum size of a maximal \( \Delta_k \)-set of \( G \) is the lower \( k \)-matching number of \( G \) and is denoted by \( a_k''(G) \). In this paper, we consider \( S_{k,m} \)-colorings of a graph and study relations between \( S_{k,m} \)-colorings and \( \Delta_k \)-sets in graphs. Bounds are established for the \( S_{k,m} \)-chromatic indexes \( \chi_{S_{k,m}}'(G) \) and \( \chi_{S_{k,m}}''(G) \) of a graph \( G \) in terms of the \( k \)-matching numbers \( a_k'(G) \) and \( a_k''(G) \) of the graph. Other results and questions are also presented.

Let \( G \) be a Hamiltonian graph of order \( n \geq 3 \). For an integer \(\ell\) with \(1 \leq \ell \leq n\), the graph \( G \) is \(\ell\)-path-Hamiltonian if every path of order \(\ell\) lies on a 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 lies on a Hamiltonian cycle of \( G \). For an integer \(\ell\) with \(2 \leq \ell \leq n-1\), the graph \( G \) is \(\ell\)-path-pancyclic if every path of order \(\ell\) in \( G \) lies on a cycle of every length from \(\ell+1\) to \(n\). (Thus, a \(2\)-path-pancyclic graph is edge-pancyclic.) A graph \( G \) of order \( n \geq 3 \) is path-pancyclic if \( G \) is \(\ell\)-path-pancyclic for each integer \(\ell\) with \(2 \leq \ell \leq n-1\). In this paper, we present a brief survey of known results on these two parameters and investigate the \(\ell\)-path-Hamiltonian graphs and \(\ell\)-path-pancyclic graphs having small minimum degree and small values of \(\ell\). Furthermore, highly path-pancyclic graphs are characterized and several well-known classes of \(2\)-path-pancyclic graphs are determined. The relationship among these two parameters and other well-known Hamiltonian parameters is investigated along with some open questions in this area of research.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). A \((p, q)\)-graph \( G = (V, E) \) is said to be AL(\(k\))-traversal if there exists a sequence of vertices \((v_1, v_2, \ldots, v_p)\) such that for each \( i = 1, 2, \ldots, p-1 \), the distance between \( v_i \) and \( v_{i+1} \) is equal to \( k \). We call a graph \( G \) a 2-steps Hamiltonian graph if it has an AL(2)-traversal in \( G \) and \( d(v_p, v_1) = 2 \). In this paper, we characterize some cubic graphs that are 2-steps Hamiltonian. We show that no forbidden subgraph characterization for non-2-steps-Hamiltonian cubic graphs is available by demonstrating that every cubic graph is a homeomorphic subgraph of a non-2-steps Hamiltonian cubic graph.

For a graph \(G = (V, E)\) and a coloring \(f : V(G) \to \mathbb{Z}_2\), let \(v_f(i) = |f^{-1}(i)|\). \(f\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). The coloring \(f : V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f_+ : E(G) \to \mathbb{Z}_2\) defined by \(f_+(xy) = |f(x) – f(y)|\), for all \(xy \in E(G)\). Let \(e_f(i) = |f_+^{-1}(i)|\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined by
\[
FI(G) = \{ |e_f(1) – e_f(0)| : f \text{ is a friendly vertex labeling of } G \}.
\]

In this paper, we determine the friendly index set of certain classes of trees and introduce a few classes of fully cordial trees.

Let \( G = (V(G), E(G)) \) be a simple, finite, and undirected graph with \( n \) vertices. Given a bijection \( f : V(G) \to \{1, \dots, n\} \), one can associate two integers \( S = f(u) + f(v) \) and \( D = |f(u) – f(v)| \) with every edge \( uv \in E(G) \). The labeling \( f \) induces an edge labeling \( f’ : E(G) \to \{0, 1\} \) such that for any edge \( uv \) in \( E(G) \), \( f'(uv) = 1 \) if \(\gcd(S, D) = 1\), and \( f'(uv) = 0 \) otherwise. Such a labeling is called an SD-prime labeling if \( f'(uv) = 1 \) for all \( uv \in E(G) \). We say that \( G \) is SD-prime if it admits an SD-prime labeling. A graph \( G \) is said to be a \emph{strongly SD-prime graph} if for every vertex \( v \) of \( G \) there exists an SD-prime labeling \( f \) satisfying \( f(v) = 1 \). In this paper, we first give some sufficient conditions for a theta graph to be strongly SD-prime. We then provide constructions of new SD-prime graphs from known SD-prime graphs and investigate the SD-primality of some general graphs.

Recently, the authors proposed a fundamental theorem for the decomposing of a complete bipartite graph. They applied the theorem to obtain complete results on the decomposition of a complete bipartite graph into connected subgraphs on four vertices and up to four edges. In this paper, we decompose a complete multi-bipartite graph into its subgraphs of four vertices and five edges. We show that necessary conditions are sufficient for the decompositions, with some exceptions where decompositions do not exist

The authors previously defined the Stanton-type graph \(S(n,m)\) and demonstrated how to decompose \(\lambda K_n\) (for the appropriate minimal values of \(\lambda\)) into Stanton-type graphs \(S(4, 3)\) of the LOE-, OLE-, LEO-, and ELO-types. Sarvate and Zhang showed that for all possible values of \(\lambda\), the necessary conditions are sufficient for LOE- and OLE-decompositions. In this paper, we show that for all possible values of \(\lambda\), the necessary conditions are sufficient for LEO- and ELO-decompositions.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A \((p, q)\)-graph \(G = (V, E)\) is said to be \(AL(k)\)-traversal if there exists a sequence of vertices \(\{v_1, v_2, \ldots, v_p\}\) such that for each \(i = 1, 2, \ldots, p-1\), the distance between \(v_i\) and \(v_{i+1}\) is equal to \(k\). We call a graph \(G\) a \(k\)-steps Hamiltonian graph if it has an \(AL(k)\)-traversal in \(G\) and the distance between \(v_p\) and \(v_1\) is \(k\). In this paper, we completely classify whether a subdivision graph of a cycle with a chord is \(2\)-steps Hamiltonian.

A triple system is decomposable if the blocks can be partitioned into two sets, each of which is itself a triple system. It is cyclically decomposable if the resulting triple systems are themselves cyclic. In this paper, we prove that a cyclic two-fold triple system is cyclically indecomposable if and only if it is indecomposable. Moreover, we construct cyclic three-fold triple systems of order $v$ which are cyclically indecomposable but decomposable for all \(v \equiv 3 \mod 6\). The only known example of a cyclic three-fold triple system of order \(v \equiv 1 \mod 6\) that is cyclically indecomposable but decomposable was a triple system on 19 points. We present a construction which yields infinitely many such triple systems of order \(v \equiv 1 \mod 6\). We also give several examples of cyclically indecomposable but decomposable cyclic four-fold triple systems and few constructions that yield infinitely many such triple systems.

The spectrum problem for decomposition of trees with up to eight edges was introduced and solved in 1978 by Huang and Rosa. Additionally, the packing problem was settled for all trees with up to six edges by Roditty. For the first time, we consider obtaining all possible leaves in a maximum tree-packing of \(K_n\), which we refer to as the spectrum problem for packings of complete graphs. In particular, we completely solve this problem for trees with at most five edges. The packing designs are used in developing optimal error-correcting codes, which have applications in biology, such as in DNA sequencing.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A labeling \(f\) of a graph \(G\) is said to be edge-friendly if \(|e_f(0) – e_f(1)| \leq 1\), where \(e_f(i) = \text{card}\{e \in E(G) : f(e) = i\}\). An edge-friendly labeling \(f : E(G) \to \mathbb{Z}_2\) induces a partial vertex labeling \(f^+ : V(G) \to A\) defined by \(f^+(x) = 0\) if the edges incident to \(x\) are labeled \(0\) more than \(1\). Similarly, \(f^+(x) = 1\) if the edges incident to \(x\) are labeled \(1\) more than \(0\). \(f^+(x)\) is not defined if the edges incident to \(x\) are labeled \(1\) and \(0\) equally. The edge-balance index set of the graph \(G\), \(EBI(G)\), is defined as \(\{|v_f(0) – v_f(1)| : \text{the edge labeling } f \text{ is edge-friendly}\}\), where \(v_f(i) = \text{card}\{v \in V(G) : f^+(v) = i\}\).

An \(n\)-wheel is a graph consisting of \(n\) cycles, with each vertex of the cycles connected to one central hub vertex. The edge-balance index sets of \(n\)-wheels are presented.

A set of vertices \(W\) \emph{locally resolves} a graph \(G\) if every pair of adjacent vertices is uniquely determined by its coordinate of distances to the vertices in \(W\). The minimum cardinality of a local resolving set of \(G\) is called the \emph{local metric dimension} of \(G\). A graph \(G\) is called a \(k\)-regular graph if every vertex of \(G\) is adjacent to \(k\) other vertices of \(G\). In this paper, we determine the local metric dimension of an \((n-3)\)-regular graph \(G\) of order \(n\), where \(n \geq 5\).

Let \(\mathcal{G}_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T: \mathcal{G}_n \to \mathcal{G}_n\), such that the dot product dimension of \(T(G)\) is the same as the dot product dimension of \(G\) for any \(G \in \mathcal{G}_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings that preserve sets of graphs with specified dot product dimensions.

A permutation \(\pi\) on a set of positive integers \(\{a_1, a_2, \ldots, a_n\}\) is said to be graphical if there exists a graph containing exactly \(a_i\) vertices of degree \(\pi(a_i)\) for each \(i\) (\(1 \leq i \leq n\)). It has been shown that for positive integers with \(a_1 < a_2 < \ldots < a_n\), if \(\pi(a_n) = a_n\), then the permutation \(\pi\) is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i)\).

We use a criterion of Tripathi and Vijay to provide a new proof of this result and to establish a similar result for permutations \(\pi\) such that \(\pi(a_{n-1}) = a_n\). We prove that such a permutation is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_na_{n-1} \leq a_{n-1}(a_{n-1} – 1) + \sum_{i\neq n-1} a_i\pi(a_i)\). We also consider permutations such that \(\pi(a_n) = a_{n-1}\) and, more generally, those such that \(\pi(a_n) = a_{n-j}\) for some \(j\) (\(1 < j < n\)).

A \(k\)-labeling of a graph is a labeling of the vertices of the graph by \(k\)-tuples of non-negative integers such that two vertices of \(G\) are adjacent if and only if their label \(k\)-tuples differ in each coordinate. The dimension of a graph \(G\) is the least \(k\) such that \(G\) has a \(k\)-labeling.

Lovász et al. showed that for \(n \geq 3\), the dimension of a path of length \(n\) is \((\log_2 n)^+\). Lovász et al. and Evans et al. determined the dimension of a cycle of length \(n\) for most values of \(n\). In the present paper, we obtain the dimension of a caterpillar or provide close bounds for it in various cases.