A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

The hub cover pebbling number, \(h^{*}(G)\), of a graph $G$, is the least non-negative integer such that from all distributions of \(h^{*}(G)\) pebbles over the vertices of \(G\), it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for \(G\) using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.

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.

A \((p, g)\)-graph \(G\) is Euclidean if there exists a bijection \(f: V \to \{1, 2, \ldots, p\}\) such that for any induced \(C_3\)-subgraph \(\{v_1, v_2, v_3\}\) in \(G\) with \(f(v_1) < f(v_2) < f(v_3)\), we have that \(f(v_1) + f(v_2) > f(v_3)\). The Euclidean Deficiency of a graph \(G\) is the smallest integer \(k\) such that \(G \cup N_k\) is Euclidean. We study the Euclidean Deficiency of one-point union and one-edge union of complete graphs.

The dominating set of a graph \(G\) is a set of vertices \(D\) such that for every \(v \in V(G)\) either \(v \in D\) or \(v\) is adjacent to a vertex in \(D\). The domination number, denoted \(\gamma(G)\), is the minimum number of vertices in a dominating set. In 1998, Haynes and Slater [1] introduced paired-domination. Building on paired-domination, we introduce 3-path domination. We define a 3-path dominating set of \(G\) to be \(D = \{ Q_1,Q_2,\dots , Q_k\, |\:Q_i \text{ is a 3-path}\}\) such that the vertex set \(V(D) = V(Q_1) \cup V(Q_2) \cup \dots \cup V(Q_k)\) is a dominating set. We define the 3-path domination number, denoted by \(\gamma_{P_3}(G)\), to be the minimum number of 3-paths needed to dominate \(G\). We show that the 3-path domination problem is NP-complete. We also prove bounds on \(\gamma_{P_3}(G)\) and improve those bounds for particular families of graphs such as Harary graphs, Hamiltonian graphs, and subclasses of trees. In general, we prove \(\gamma_{P_3}(G) \leq \frac{n}{3}\).

Two colorings have been introduced recently where an unrestricted coloring \(c\) assigns nonempty subsets of \([k]=\{1,\ldots,k\}\) to the edges of a (connected) graph \(G\) and gives rise to a vertex-distinguishing vertex coloring by means of set operations. If each vertex color is obtained from the union of the incident edge colors, then \(c\) is referred to as a strong royal coloring. If each vertex color is obtained from the intersection of the incident edge colors, then \(c\) is referred to as a strong regal coloring. The minimum values of \(k\) for which a graph \(G\) has such colorings are referred to as the strong royal index of \(G\) and the strong regal index of \(G\) respectively. If the induced vertex coloring is neighbor distinguishing, then we refer to such edge colorings as royal and regal colorings. The royal chromatic number of a graph involves minimizing the number of vertex colors in an induced vertex coloring obtained from a royal coloring. In this paper, we provide new results related to these two coloring concepts and establish a connection between the corresponding chromatic parameters. In addition, we establish the royal chromatic number for paths and cycles.