Unlike undirected graphs where the concept of Roman domination has been studied very extensively over the past 15 years, Roman domination remains little studied in digraphs. However, the published works are quite varied and deal with different aspects of Roman domination, including for example, Roman bondage, Roman reinforcement, Roman dominating family of functions and the signed version of some Roman dominating functions. In this survey, we will explore some Roman domination related results on digraphs, some of which are extensions of well-known properties of the Roman domination parameters of undirected graphs.

Let \( K_{g_1,g_2,\dots,g_n} \) be a complete \( n \)-partite graph with partite sets of sizes \( g_i \) for \( 1 \leq i \leq n \). A complete \( n \)-partite is balanced if each partite set has \( g \) vertices. In this paper, we will solve the problem of finding a maximum packing of the balanced complete \( n \)-partite graph, \( n \) even, with edge-disjoint 5-cycles when the leave is a 1-factor.

A double Italian dominating function on a graph \( G \) with vertex set \( V(G) \) is defined as a function \( f : V(G) \to \{0,1,2,3\} \) such that each vertex \( u \in V(G) \) with \( f(u) \in \{0,1\} \) has the property that \( \sum_{x \in N[u]} f(x) \geq 3 \), where \( N[u] \) is the closed neighborhood of \( u \). A set \( \{f_1, f_2, \dots, f_d\} \) of distinct double Italian dominating functions on \( G \) with the property that \( \sum_{i=1}^{d} f_i(v) \leq 3 \) for each \( v \in V(G) \) is called a \textit{double Italian dominating family} (of functions) on \( G \). The maximum number of functions in a double Italian dominating family on \( G \) is the double Italian domatic number of \( G \), denoted by \( dd_I(G) \). We initiate the study of the double Italian domatic number, and we present different sharp bounds on \( dd_I(G) \). In addition, we determine the double Italian domatic number of some classes of graphs.

We define the push statistic on permutations and multipermutations and use this to obtain various results measuring the degree to which an arbitrary permutation deviates from sorted order. We study the distribution on permutations for the statistic recording the length of the longest push and derive an explicit expression for its first moment and generating function. Several auxiliary concepts are also investigated. These include the number of cells that are not pushed; the number of cells that coincide before and after pushing (i.e., the fixed cells of a permutation); and finally the number of groups of adjacent columns of the same height that must be reordered at some point during the pushing process.

Let \( \mathcal{K} \) be a family of sets in \( \mathbb{R}^d \). For each countable subfamily \( \{K_m : m \geq 1\} \) of \( \mathcal{K} \), assume that \( \bigcap \{K_m : m \geq 1\} \) is consistent relative to staircase paths and starshaped via staircase paths, with a staircase kernel that contains a convex set of dimension \( d \). Then \( \bigcap \{K : K \in \mathcal{K} \} \) has these properties as well. For \( n \) fixed, \( n \geq 1 \), an analogous result holds for sets starshaped via staircase \( n \)-paths.

We introduce a new bivariate polynomial which we call the defensive alliance polynomial and denote it by \( da(G; x, y) \). It is a generalization of the alliance polynomial [Carballosa et al., 2014] and the strong alliance polynomial [Carballosa et al., 2016]. We show the relation between \( da(G; x, y) \) and the alliance, the strong alliance, and the induced connected subgraph [Tittmann et al., 2011] polynomials. Then, we investigate information encoded in \( da(G; x, y) \) about \( G \). We discuss the defensive alliance polynomial for the path graphs, the cycle graphs, the star graphs, the double star graphs, the complete graphs, the complete bipartite graphs, the regular graphs, the wheel graphs, the open wheel graphs, the friendship graphs, the triangular book graphs, and the quadrilateral book graphs. Also, we prove that the above classes of graphs are characterized by its defensive alliance polynomial. A relation between induced subgraphs with order three and both subgraphs with order three and size three and two respectively, is proved to characterize the complete bipartite graphs. Finally, we present the defensive alliance polynomial of the graph formed by attaching a vertex to a complete graph. We show two pairs of graphs which are not characterized by the alliance polynomial but characterized by the defensive alliance polynomial.

A cancellable number (CN) is a fraction in which a decimal digit can be removed (“canceled”) in the numerator and denominator without changing the value of the number; examples include \( \frac{64}{16} \) where the 6 can be canceled and \( \frac{98}{49} \) where the 9 can be canceled. We present a few limit theorems and provide several generalizations.