An \( H \)-decomposition of a graph \( G \) is a partition of the edges of \( G \) into copies isomorphic to \( H \). When the decomposition is not feasible, one looks for the best possible by minimizing: the number of unused edges (leave of a packing), or the number of reused edges (padding of a covering). We consider the \( H \)-decomposition, packing, and covering of the complete graphs and complete bipartite graphs, where \( H \) is a 4-cycle with three pendant edges.

We introduce a new bivariate polynomial
\[
J(G; x, y) := \sum_{W \subseteq V(G)} x^{|W|} y^{|N[W]| – |W|}
\]
which contains the standard domination polynomial of the graph \( G \) in two different ways. We build methods for efficient calculation of this polynomial and prove that there are still some families of graphs which have the same bivariate polynomial.

Let \( G \) be a \( (p, q) \) graph. Let \( f : V(G) \to \{1, 2, \ldots, k\} \) be a map where \( k \) is an integer \( 2 \leq k \leq p \). For each edge \( uv \), assign the label \( |f(u) – f(v)| \). \( f \) is called \( k \)-difference cordial labeling of \( G \) if \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(0) – e_f(1)| \leq 1 \), where \( v_f(x) \) denotes the number of vertices labeled with \( x \), \( e_f(1) \) and \( e_f(0) \) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a \( k \)-difference cordial labeling is called a \( k \)-difference cordial graph. In this paper, we investigate 3-difference cordial labeling behavior of slanting ladder, book with triangular pages, middle graph of a path, shadow graph of a path, triangular ladder, and the armed crown.

In this paper, we consider the sequences \( \{F(n, k)\}_{n \geq k} \) (\(k \geq 1\)) defined by\( F(n, k) = (n – 2)F(n – 1, k) + F(n – 1, k – 1), \quad F(n, 1) = \frac{n!}{2}, \quad F(n, n) = 1. \) We mainly study the log-convexity of \( \{F(n, k)\}{n \geq k} \) (\(k \geq 1\)) when \( k \) is fixed. We prove that \( \{F(n, 3)\}{n \geq 3}, \{F(n, 4)\}{n \geq 5}, \) and \( \{F(n, 5)\}{n \geq 6} \) are log-convex. In addition, we discuss the log-behavior of some sequences related to \( F(n, k) \).
\end{abstract}

Let \( G = C_n \oplus C_n \) with \( n \geq 3 \) and \( S \) be a sequence with elements of \( G \). Let \( \Sigma(S) \subseteq G \) denote the set of group elements which can be expressed as a sum of a nonempty subsequence of \( S \). In this note, we show that if \( S \) contains \( 2n – 3 \) elements of \( G \), then either \( 0 \in \Sigma(S) \) or \( |\Sigma(S)| \geq n^2 – n – 1 \). Moreover, we determine the structures of the sequence \( S \) over \( G \) with length \( |S| = 2n – 3 \) such that \( 0 \notin \Sigma(S) \) and \( |\Sigma(S)| = n^2 – n – 1 \).

Let \(G\) be a finite and simple graph with vertex set \(V(G)\). A nonnegative signed Roman dominating function (NNSRDF) on a graph \(G\) is a function \(f:V(G)\to \{-1,1,2\}\) satisfying the conditions that (i) \(\sum_{x\in N[v]}f(x)\ge 0\) for each \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v\) and (ii)every vertex u for which \(f(u)=-1\) has a neighbor v for which \(f(v)=2\). The weight of an NNSRDF \(f\) is \(\omega(f) = \sum_{v\in V(G)} f(v)\). The nonnegative signed Roman domination number \(\gamma_{sR}^{NN} (G)\) of \(G\) is the minimum weight of an NNSRDF \(G\) In this paper, we initiate the study of the nonnegative signed Roman domination number of a graph and we present different bounds on \(\gamma _{sR}^{NN}(G) \ge (8n-12m)/7\). In addition, if \(G\) is a bipartite graph of order \(n\), then we prove that \(\gamma _{sR}^{NN}(G) \ge^\frac{3}{2}(\sqrt{4n+9}-3)-n\), and we characterize the external graphs.

We consider inverse-conjugate compositions of a positive integer \(n\) in which the parts belong to the residue class of 1 modulo an integer \(m > 0\). It is proved that such compositions exist only for values of \(n\) that belong to the residue class of 1 modulo 2m. An enumerations results is provided using the properties of inverse-conjugate compositions. This work extends recent results for inverse-conjugate compositions with odd parts.

For a graph \(G\) and positive integers \(a_1,…,a_r,\) if every r-coloring of vertics V(G) must result in a monochromatic \(a_1\)-clique of color \(i\) for some \(i \in \{1,…,r\},\) then we write \(G \to (a_1,..a_r)^v\).\(F_v(K_a1,…,K_ar;H)\) is the smallest integer \(n\) such that there is an H-free graph \(G\) of order \(n\), and \(G \to (a_1,…,a_r)^v\). In this paper we study upper and lower bounds for some generalized vertex Folkman numbers of from \(F_v(K_{a1},…,K_{ar};K_4 – e)\), where \(r \in {2,3}\) and \(a_1 \in {2,3}\) for 10 and \(F_v(K_2,K_3;K_4 – e) = 19\) by computing, and prove \(F_v(K_3,K_3;K_4 – e)\ge F_v(K_2,K_2,K_3;K_4 – e)\ge 25\)

We study the complexity of four decision problems dealing with the uniqueness of a solution in a graph: “Uniqueness of a Vertex Cover with bounded size” (U-VC) and “Uniqueness of an Optimal Vertex Cover” (U-OVC), and for any fixed integer \(r \ge 1,\) “Uniqueness of an \(r\)-Dominating Code with bounded size” \((U-DC_r)\) and “Uniqueness of an Optimal \(r\)-Dominating Code” \((U-ODC_r\). In particular, we give a polynomial reduction from “Unique Satisfiability of a Boolean formula” (U-SAT) to U-OVC, and from U-SAT to U-ODC, We prove that U-VC and \(U-DC_r\) have complexity equivalent to that of U-SAT (up to polynomials); consequently, these problems are all \(NP\)-hard, and U-VC and \(U-DC_r\) belong to the class \(DP\).

Let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A signed total Roman dominating function on the digraph \(D\) is a function \(f : V(D)\longrightarrow{-1,1,2}\) \(\sum_{u\in N-(v)} f(u)\ge 1\) for every \(v\in V(D)\), where \(N^{-}(v)\) consists of all inner neighbors of \(v\) for dominating function on \(D\) with the property that \(\sum_{d}^{i=1}f_i(v)\le 1\) for each \(v \in V (D)\) is called a signed total roman dominating family (of functions) on \(D\). The maximum number of functions in a signed total roman dominating family on \(D\)is the signed total Roman domatic number of \(D\). denoted by \(d_{stR}(D)\). In addition, we determine the signed total Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the signed total Roman domatic of graphs.