An outer independent double Roman dominating function (OIDRDF) on a graph \( G \) is a function \( f : V(G) \to \{0, 1, 2, 3\} \) having the property that (i) if \( f(v) = 0 \), then the vertex \( v \) must have at least two neighbors assigned 2 under \( f \) or one neighbor \( w \) with \( f(w) = 3 \), and if \( f(v) = 1 \), then the vertex \( v \) must have at least one neighbor \( w \) with \( f(w) \ge 2 \) and (ii) the subgraph induced by the vertices assigned 0 under \( f \) is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \( \gamma_{oidR}(G) \) is the minimum weight of an OIDRDF on \( G \). The \( \gamma_{oidR} \)-stability (\( \gamma^-_{oidR} \)-stability, \( \gamma^+_{oidR} \)-stability) of \( G \), denoted by \( {\rm st}_{\gamma_{oidR}}(G) \) (\( {\rm st}^-_{\gamma_{oidR}}(G) \), \( {\rm st}^+_{\gamma_{oidR}}(G) \)), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the \( \gamma_{oidR} \)-stability of some special classes of graphs, and present some bounds on \( {\rm st}_{\gamma_{oidR}}(G) \). In addition, for a tree \( T \) with maximum degree \( \Delta \), we show that \( {\rm st}_{\gamma_{oidR}}(T) = 1 \) and \( {\rm st}^-_{\gamma_{oidR}}(T) \le \Delta \), and characterize the trees that achieve the upper bound.

We introduce a two-player game where the goal is to illuminate all edges of a graph. At each step the first player, called Illuminator, taps a vertex. The second player, called Adversary, reveals the edges incident with that vertex (consistent with the edges incident with the already tapped vertices). Illuminator tries to minimize the taps needed, and the value of the game is the number of taps needed with optimal play. We provide bounds on the value in trees and general graphs. In particular, we show that the value for the path on \( n \) vertices is \( \frac{2}{3} n + O(1) \), and there is a constant \( \varepsilon > 0 \) such that for every caterpillar on \( n \) vertices, the value is at most \( (1 – \varepsilon) n + 1 \).

Let \(G\) be a group, and let \(c\in\mathbb{Z}^+\cup\{\infty\}\). We let \(\sigma_c(G)\) be the maximal size of a subset \(X\) of \(G\) such that, for any distinct \(x_1,x_2\in X\), the group \(\langle x_1,x_2\rangle\) is not \(c\)-nilpotent; similarly we let \(\Sigma_c(G)\) be the smallest number of \(c\)-nilpotent subgroups of \(G\) whose union is equal to \(G\). In this note we study \(D_{2k}\), the dihedral group of order \(2k\). We calculate \(\sigma_c(D_{2k})\) and \(\Sigma_c(D_{2k})\), and we show that these two numbers coincide for any given \(c\) and \(k\).

Let \(p > 2\) be prime and \(r \in \{1,2, \ldots, p-1\}\). Denote by \(c_{p}(n)\) the number of \(p\)-regular partitions of \(n\) in which parts can occur not more than three times. We prove the following: If \(8r + 1\) is a quadratic non-residue modulo \(p\), \(c_{p}(pn + r) \equiv 0 \pmod{2}\) for all nonnegative integers \(n\).

Let \( G=(V,E) \) be a simple connected graph with vertex set \( G \) and edge set \( E \). The harmonic index of graph \( G \) is the value \( H(G)=\sum_{uv\in E(G)} \frac{2}{d_u+d_v} \), where \( d_x \) refers to the degree of \( x \). We obtain an upper bound for the harmonic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound.

One of the fundamental properties of the hypercube \( Q_n \) is that it is bipancyclic as \( Q_n \) has a cycle of length \( l \) for every even integer \( l \) with \( 4 \leq l \leq 2^n \). We consider the following problem of generalizing this property: For a given integer \( k \) with \( 3 \leq k \leq n \), determine all integers \( l \) for which there exists an \( l \)-vertex, \( k \)-regular subgraph of \( Q_n \) that is both \( k \)-connected and bipancyclic. The solution to this problem is known for \( k = 3 \) and \( k = 4 \). In this paper, we solve the problem for \( k = 5 \). We prove that \( Q_n \) contains a \( 5 \)-regular subgraph on \( l \) vertices that is both \( 5 \)-connected and bipancyclic if and only if \( l \in \{32, 48\} \) or \( l \) is an even integer satisfying \( 52 \leq l \leq 2^n \). For general \( k \), we establish that every \( k \)-regular subgraph of \( Q_n \) has \( 2^k, 2^k + 2^{k-1} \) or at least \( 2^k + 2^{k-1} + 2^{k-3} \) vertices.

Coded caching technology can better alleviate network traffic congestion. Since many of the centralized coded caching schemes now in use have high subpacketization, which makes scheme implementation more challenging, coded caching schemes with low subpacketization offer a wider range of practical applications. It has been demonstrated that the coded caching scheme can be achieved by creating a combinatorial structure named placement delivery array (PDA). In this work, we employ vector space over a finite field to obtain a class of PDA, calculate its parameters, and consequently achieve a coded caching scheme with low subpacketization. Subsequently, we acquire a new MN scheme and compare it with the new scheme developed in this study. The subpacketization \(F\) of the new scheme has significant advantages. Lastly, the number of users \(K\), cache fraction \(\frac{M}{N}\), and subpacketization \(F\) have advantages to some extent at the expense of partial transmission rate \(R\) when compared to the coded caching scheme in other articles.

We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with \(C_{3}\) and \(C_{4}\) vanes, and infinite families of \(3,5\)-windmills and \(3,6\)-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of \(t\) 5-cycles with one vertex in common (\(C_{5}^{t}\)) is graceful if and only if \(t \equiv 0, 3 \pmod{4}\) and near graceful when \(t \equiv 1, 2 \pmod{4}\).

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence \((G_n)_{n \geq 0}\), where \(G_n\) is the group generated by a subset of the symmetric group \(S_n\) consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence \((G_n)_{n \geq 0}\) is eventually constant. Moreover, we find a set of patterns such that \((G_n)_{n \geq 0}\) is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.