Let \( G \) be a simple and finite graph. A graph is said to be decomposed into subgraphs \( H_1 \) and \( H_2 \) which is denoted by \(G = H_1 \oplus H_2,\) if \( G \) is the edge-disjoint union of \( H_1 \) and \( H_2 \). If \(G = H_1 \oplus H_2 \oplus \dots \oplus H_k,
\)where \( H_1, H_2, \dots, H_k \) are all isomorphic to \( H \), then \( G \) is said to be \( H \)-decomposable. Furthermore, if \( H \) is a cycle of length \( m \), then we say that \( G \) is \( C_m \)-decomposable and this can be written as \( C_m \mid G \). Where \( G \times H \) denotes the tensor product of graphs \( G \) and \( H \), in this paper, we prove that the necessary conditions for the existence of \( C_6 \)-decomposition of \( K_m \times K_n \) are sufficient. Using these conditions, it can be shown that every even regular complete multipartite graph \( G \) is \( C_6 \)-decomposable if the number of edges of \( G \) is divisible by 6.

Constructions of the smallest known trivalent graph for girths 17, 18, and 20 are given. All three graphs are voltage graphs. Their orders are 2176, 2560, and 5376, respectively, improving the previous values of 2408, 2640, and 6048.

A signed total Italian dominating function (STIDF) of a graph \( G \) with vertex set \( V(G) \) is defined as a function \( f : V(G) \to \{-1,1,2\} \) having the property that (i) \( \sum_{x \in N(v)} f(x) \geq 1 \) for each \( v \in V(G) \), where \( N(v) \) is the neighborhood of \( v \), and (ii) every vertex \( u \) for which \( f(u) = -1 \) is adjacent to a vertex \( v \) for which \( f(v) = 2 \) or adjacent to two vertices \( w \) and \( z \) with \( f(w) = f(z) = 1 \). The weight of an STIDF is the sum of its function values over all vertices. The \textit{signed total Italian domination number} of \( G \), denoted by \( \gamma_{stI}(G) \), is the minimum weight of an STIDF in \( G \). We initiate the study of the signed total Italian domination number, and we present different sharp bounds on \( \gamma_{stI}(G) \). In addition, we determine the signed total Italian domination number of some classes of graphs.

One may generalize integer compositions by replacing positive integers with elements from an additive group, giving the broader concept of compositions over a group. In this note, we give some simple bijections between compositions over a finite group. It follows from these bijections that the number of compositions of a nonzero group element \( g \) is independent of \( g \). As a result, we derive a simple expression for the number of compositions of any given group element. This extends an earlier result for abelian groups which was obtained using generating functions and a multivariate multisection formula.

An acyclic ordering of a directed acyclic graph (DAG) \( G \) is a sequence \( \alpha \) of the vertices of \( G \) with the property that if \( i < j \), then there is a path in \( G \) from \( \alpha(i) \) to \( \alpha(j) \). In this paper, we explore the problem of finding the number of possible acyclic orderings of a general DAG. The main result is a method for reducing a general DAG to a simpler one when counting the number of acyclic orderings. Along the way, we develop a formula for quickly obtaining this count when a DAG is a tree.

The diagnosability of a multiprocessor system is one important study topic. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the \( g \)-extra diagnosability, which restrains that every fault-free component has at least \( (g+1) \) fault-free nodes. As a favorable topology structure of interconnection networks, the \( n \)-dimensional alternating group graph \( AG_n \) has many good properties. In this paper, we prove that the 3-extra diagnosability of \( AG_n \) is \( 8n – 25 \) for \( n \geq 5 \) under the PMC model and for \( n \geq 7 \) MM* model.

M. Klešc et al. characterized graphs \( G_1 \) and \( G_2 \) for which the crossing number of their Cartesian product \( G_1 \square G_2 \) equals one or two. In this paper, their results are extended by giving the necessary and sufficient conditions for all pairs of graphs \( G_1 \) and \( G_2 \) for which the crossing number of their Cartesian product \( G_1 \square G_2 \) equals three, if one of the graphs \( G_1 \) and \( G_2 \) is a cycle.

In this paper we study relations between connected and weakly convex domination numbers. We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number. We also study the influence of the edge removing on the weakly convex domination number, in particular we prove that the weakly convex domination number is an interpolating function.

Denote by \( p_m \) the \( m \)-th prime number (\( p_1 = 2 \), \( p_2 = 3 \), \( p_3 = 5 \), \( p_4 = 7 \), \dots). Let \( T \) be a rooted tree with branches \( T_1, T_2, \dots, T_r \). The Matula number \( M(T) \) of \( T \) is \( p_{M(T_1)} \cdot p_{M(T_2)} \cdots p_{M(T_r)} \), starting with \( M(K_1) = 1 \). This number was put forward half a century ago by the American mathematician David Matula. In this paper, we prove that the star (consisting of a root and leaves attached to it) and the binary caterpillar (a binary tree whose internal vertices form a path starting at the root) have the smallest and greatest Matula number, respectively, over all topological trees (rooted trees without vertices of outdegree 1) with a prescribed number of leaves – the extreme values are also derived.