The stretched Littlewood-Richardson coefficient $c_{t\lambda ,t\mu }^{t\nu }$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t$. It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg’s formula, the hive conditions, and the Kostant partition function to show a stronger result that $c_{\lambda ,\mu }^{\nu }$ is indeed a polynomial in variables $\nu ,\lambda ,\mu$ provided they lie in certain polyhedral cones. Motivated by Rassart’s approach, we give a short alternative proof of the polynomiality of $c_{t\lambda ,t\mu }^{t\nu }$ using Steinberg’s formula and a simple argument about the chamber complex of the Kostant partition function.

In this work, we study type B set partitions for a given specific positive integer $k$ defined over $⟨n⟩=\{-n,-(n-1),\cdots ,-1,0,1,\cdots ,n-1,n\}$. We found a few generating functions of type B analogues for some of the set partition statistics defined by Wachs, White and Steingrímsson for partitions over positive integers $[n]=\{1,2,\cdots ,n\}$, both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over $[n]$, in the scenario of $⟨n⟩$ and called the analogue as Signed Restricted Growth Function (SRGF). We discussed analogues of major index for type B partitions in terms of SRGF. We found an analogue of Foata bijection and reduced matrix for type B set partitions as done by Sagan for set partitions of $[n]$ with specific number of blocks $k$. We conclude with some open questions regarding the type B analogue of some well known results already done in case of set partitions of $[n]$.

Suppose that $\varphi$ is a proper edge-$k$-coloring of the graph $G$. For a vertex $v\in V(G)$, let $C_{\varphi }(v)$ denote the set of colors assigned to the edges incident with $v$. The proper edge-$k$-coloring $\varphi$ of $G$ is strict neighbor-distinguishing if for any adjacent vertices $u$ and $v$, $C_{\varphi }(u)⊊C_{\varphi }(v)$ and $C_{\varphi }(v)⊊C_{\varphi }(u)$. The strict neighbor-distinguishing index, denoted ${\chi }_{snd}^{\prime }(G)$, is the minimum integer $k$ such that $G$ has a strict neighbor-distinguishing edge-$k$-coloring. In this paper we prove that if $G$ is a simple graph with maximum degree five, then ${\chi }_{snd}^{\prime }(G)\le 12$.

Let $2\le k\in \mathbb{Z}$. A total coloring of a $k$-regular simple graph via $k+1$ colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. In this work, focus is set upon graphs of girth $k+1$. Efficient total colorings of finite connected simple cubic graphs of girth 4 are constructed starting at the 3-cube. It is conjectured that all of them are obtained by means of four basic operations. In contrast, the Robertson 19-vertex $(4,5)$-cage, the alternate union $Pe{t}^k$ of a (Hamilton) $10k$-cycle with $k$ pentagon and $k$-pentagram 5-cycles, for $k>1$ not divisible by 5, and its double cover $Do{d}^k$, contain TCs that are nonefficient. Applications to partitions into 3-paths and 3-stars are given.

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, Müller-Gronbach, Novak-Ritter and Ullrich.

Topological indices have become an essential tool to investigate theoretical and practical problems in various scientific areas. In chemical graph theory, a significant research work, which is associated with the topological indices, is to deduce the ideal bounds and relationships between known topological indices. Mathematical development of the novel topological index is valid only if the topological index shows a good correlation with the physico-chemical properties of chemical compounds. In this article, the chemical applicability of the novel GQ and QG indices is calibrated over physico-chemical properties of 22 benzenoid hydrocarbons. The GQ and QG indices predict the physico-chemical properties of benzenoid hydrocarbons, significantly. Additionally, this work establishes some mathematical relationships between each of the GQ and QG indices and each of the graph invariants: size, degree sequences, maximum and minimum degrees, and some well-known degree-based topological indices of the graph.

In 2003, the frequency assignment problem in a cellular network motivated Even et al. to introduce a new coloring problem: Conflict-Free coloring. Inspired by this problem and by the Gardner-Bodlaender’s coloring game, in 2020, Chimelli and Dantas introduced the Conflict-Free Closed Neighborhood $k$-coloring game (CFCN $k$-coloring game). The game starts with an uncolored graph $G$, $k\ge 2$ different colors, and two players, Alice and Bob, who alternately color the vertices of $G$. Both players can start the game and respect the following legal coloring rule: for every vertex $v$, if the closed neighborhood $N[v]$ of $v$ is fully colored then there exists a color that was used only once in $N[v]$. Alice wins if she ends up with a Conflict-Free Closed Neighborhood $k$-coloring of $G$, otherwise, Bob wins if he prevents it from happening. In this paper, we introduce the game for open neighborhoods, the Conflict-Free Open Neighborhood $k$-coloring game (CFON $k$-coloring game), and study both games on graph classes determining the least number of colors needed for Alice to win the game.

This paper investigates the number of rooted biloopless nonseparable planar near-triangulations and presents some formulae for such maps with three parameters: the valency of root-face, the number of edges and the number of inner faces. All of them are almost summation-free.

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs without 4-cycles with maximum degree $\mathrm{\Delta }\ge 10$.

For a graph $G=(V,E)$ of size $q$, a bijection $f:E\to \{1,2,\dots ,q\}$ is a local antimagic labeling if it induces a vertex labeling $f^{+}:V\to \mathbb{N}$ such that $f^{+}(u)\ne f^{+}(v)$, where $f^{+}(u)$ is the sum of all the incident edge label(s) of $u$, for every edge $uv\in E(G)$. In this paper, we make use of matrices of fixed sizes to construct several families of infinitely many tripartite graphs with local antimagic chromatic number 3.

An outer independent double Roman dominating function (OIDRDF) of a graph $G$ is a function $f:V(G)\to \{0,1,2,3\}$ satisfying the following conditions:
(i) every vertex $v$ with $f(v)=0$ is adjacent to a vertex assigned 3 or at least two vertices assigned 2;
(ii) every vertex $v$ with $f(v)=1$ has a neighbor assigned 2 or 3;
(iii) no two vertices assigned 0 are adjacent.
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 $G$. Ahangar et al. [Appl. Math. Comput. 364 (2020) 124617] established that for every tree $T$ of order $n\ge 4$, ${\gamma }_{oidR}(T)\le \frac{5}{4}n$ and posed the question of whether this bound holds for all connected graphs. In this paper, we show that for a unicyclic graph $G$ of order $n$, ${\gamma }_{oidR}(G)\le \frac{5n+2}{4}$, and for a bicyclic graph, ${\gamma }_{oidR}(G)\le \frac{5n+4}{4}$. We further characterize the graphs attaining these bounds, providing a negative answer to the question posed by Ahangar et al.

Let $G$ be a $(p,q)$ graph. Let $f$ be a function from $V(G)$ to the set $\{1,2,\dots ,k\}$ where $k$ is an integer $2<k\le |V(G)|$. For each edge $uv$ assign the label $r$ where $r$ is the remainder when $f(u)$ is divided by $f(v)$ (or) $f(v)$ is divided by $f(u)$ according as $f(u)\ge f(v)$ or $f(v)\ge f(u)$. $f$ is called a $k$-remainder cordial labeling of $G$ if $|v_f(i)-v_f(j)|\le 1$, $i,j\in \{1,\dots ,k\}$ where $v_f(x)$ denote the number of vertices labeled with $x$ and $|{\eta }_e(0)-{\eta }_o(1)|\le 1$ where ${\eta }_e(0)$ and ${\eta }_o(1)$ respectively denote the number of edges labeled with even integers and number of edges labeled with odd integers. A graph with admits a $k$-remainder cordial labeling is called a $k$-remainder cordial graph. In this paper we investigate the $4$-remainder cordial labeling behavior of Prism, Crossed prism graph, Web graph, Triangular snake, $L_n\odot m{K}_1$, Durer graph, Dragon graph.

Given a connected graph $H$, its first Zagreb index $M_1(H)$ is equal to the sum of squares of the degrees of all vertices in $H$. In this paper, we give a best possible lower bound on $M_1(H)$ that guarantees $H$ is $\tau$-path-coverable and $\tau$-edge-Hamiltonian, respectively. Our research supplies a continuation of the results presented by Feng et al. (2017).

The degree of an edge $uv$ of a graph $G$ is $d_G(u)+d_G(v)-2.$ The degree associated edge reconstruction number of a graph $G$ (or dern(G)) is the minimum number of degree associated edge-deleted subgraphs that uniquely determines $G.$ Graphs whose vertices all have one of two possible degrees $d$ and $d+1$ are called $(d,d+1)$-bidegreed graphs. It was proved, in a sequence of two papers [1,17], that $dern(m{K}_{1,3})=4$ for $m>1,$ $dern(m{K}_{2,3})=dern(r{P}_3)=3$ for $m>0,r>1$ and $dern(G)=1$ or $2$ for all other bidegreed graphs $G$ except the $(d,d+1)$-bidegreed graphs in which a vertex of degree $d+1$ is adjacent to at least two vertices of degree $d.$ In this paper, we prove that $dern(G)=1$ or $2$ for this exceptional bidegreed graphs $G.$ Thus, $dern(G)\le 4$ for all bidegreed graphs $G.$

A proper total coloring of a graph $G$ such that there are at least 4 colors on those vertices and edges incident with a cycle of $G$, is called an acyclic total coloring. The acyclic total chromatic number of $G$, denoted by ${\chi }_a^{”}(G)$, is the smallest number of colors such that $G$ has an acyclic total coloring. In this article, we prove that for any graph $G$ with $\mathrm{\Delta }(G)=\mathrm{\Delta }$ which satisfies ${\chi }^{”}(G)\le A$ for some constant $A$, and for any integer $r$, $1\le r\le 2\mathrm{\Delta }$, there exists a constant $c>0$ such that if $g(G)\ge \frac{c\mathrm{\Delta }}{r}\log \frac{{\mathrm{\Delta }}^2}{r}$, then ${\chi }_a^{”}(G)\le A+r$.

We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph $G$, we are interested in $b_2(G)$, the minimum number of rounds until the information has spread to all vertices of graph $G$. We are also interested in finding $t_2(G)$, the minimum number of sources necessary so that the information spreads to all vertices of $G$ in $b_2(G)$ rounds. In addition to general results, we find $b_2(G)$ and $t_2(G)$ for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.

An hourglass ${\mathrm{\Gamma }}_0$ is the graph with degree sequence $\{4,2,2,2,2\}$. In this paper, for integers $j\ge i\ge 1$, the bull $B_{i,j}$ is the graph obtained by attaching endvertices of two disjoint paths of lengths $i,j$ to two vertices of a triangle. We show that every 3-connected $\{K_{1,3},{\mathrm{\Gamma }}_0,X\}$-free graph, where $X\in \{B_{2,12},B_{4,10},B_{6,8}\}$, is Hamilton-connected. Moreover, we give an example to show the sharpness of our result, and complete the characterization of forbidden induced bulls implying Hamilton-connectedness of a 3-connected {claw, hourglass, bull}-free graph.

Let $G=(V,E)$ be a simple connected graph with vertex set $V$ and edge set $E$. The Randić index of graph $G$ is the value $R(G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{d(u)d(v)}}$, where $d(u)$ and $d(v)$ refer to the degree of the vertices $u$ and $v$. We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.

In this paper, it is pointed out that the definition of `Fibonacci $(p,r)$-cube’ in many papers (denoted by $I{\mathrm{\Gamma }}_n^{(p,r)}$) is incorrect. The graph $I{\mathrm{\Gamma }}_n^{(p,r)}$ is not the same as the original one (denoted by $O{\mathrm{\Gamma }}_n^{(p,r)}$) introduced by Egiazarian and Astola. First, it is shown that $I{\mathrm{\Gamma }}_n^{(p,r)}$ and $O{\mathrm{\Gamma }}_n^{(p,r)}$ have different recursive structure. Then, it is proven that all the graphs $O{\mathrm{\Gamma }}_n^{(p,r)}$ are partial cubes. However, only a small part of graphs $I{\mathrm{\Gamma }}_n^{(p,r)}$ are partial cubes. It is also shown that $I{\mathrm{\Gamma }}_n^{(p,r)}$ and $O{\mathrm{\Gamma }}_n^{(p,r)}$ have different medianicity. Finally, several questions are listed for further investigation.

A $q$-total coloring of $G$ is an assignment of $q$ colors to the vertices and edges of $G$, so that adjacent or incident elements have different colors. The Total Coloring Conjecture (TCC) asserts that a total coloring of a graph $G$ has at least $\mathrm{\Delta }+1$ and at most $\mathrm{\Delta }+2$ colors. In this paper, we determine that all members of new infinite families of snarks obtained by the Kochol superposition of Goldberg and Loupekine with Blowup and Semiblowup snarks are Type~1. These results contribute to a question posed by Brinkmann, Preissmann and D. Sasaki (2015) by presenting negative evidence about the existence of Type~2 cubic graphs with girth at least 5.

In this note, we establish six Gallai theorems involving twelve minority and majority parameters. Accordingly, the complexity problems corresponding to some of these parameters are obtained.

A $k$-tree is a graph that can be formed by starting with $K_{k+1}$ and iterating the operation of making a new vertex adjacent to all the vertices of a $k$-clique of the existing graph. A structural characterization of 3-trees with diameter at most 2 is proven. This implies a corollary for planar 3-trees which leads to a description of their degree sequences.

In this paper, we present a new combinatorial characterization of Hermitian cones in $\mathrm{P}\mathrm{G}(3,q^2)$.

In this paper we consider some new weighted and alternating weighted generalized Fibonomial sums and the corresponding $q-$forms. A generalized form of weight sequences which contains squares in subscripts is discussed for the first time in the literature. The main key to get success in sums is an ability to change one sum into another that is simpler in some way. Thus, in order to prove these sums by doing some manipulations and tricks, our approach is to use classical $q-$analysis, in particular a formula of Rothe, a version of Cauchy binomial theorem and Gauss identity.

A new series of four-associate class partially balanced incomplete block designs in two replications has been proposed. The blocks of these designs are of two different sizes. The blocks can be divided into two groups such that every treatment appears in each group exactly once, and any two blocks belonging to two different groups have a constant number of treatments in common, i.e., these designs are affine resolvable.

Let $0<k\in \mathbb{Z}$. Let the star 2-set transposition graph $S{T}_k^2$ be the $(2k-1)$-regular graph whose vertices are the $2k$-strings on $k$ symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such $2k$-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph $P{C}_k^2$ has the same vertex set of $S{T}_k^2$ and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of $S{T}_k^2$. For $1<k\in \mathbb{Z}$, we show that $S{T}_k^2$ and $P{C}_k^2$, among other intermediate transposition graphs, have total colorings via $2k-1$ colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of $S{T}_k^2$ and $P{C}_k^2$, and partitions into $2k-1$ such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.

This paper investigates the Turan-like problem for ${\mathcal{K}}_{r+1}^{-}$-free $(r\ge 2)$ unbalanced signed graphs, where ${\mathcal{K}}_{r+1}^{-}$ is the set of unbalanced signed complete graphs with $r+1$ vertices. The maximum number of edges and the maximum index for ${\mathcal{K}}_{r+1}^{-}$-free unbalanced signed graphs are given. Moreover, the extremal ${\mathcal{K}}_{r+1}^{-}$-free unbalanced signed graphs with the maximum index are characterized.

In this paper, we give a classification of all Mengerian $4$-uniform hypergraphs derived from graphs.

The $n$-dimensional Möbius cube $M{Q}_n$ is an important variant of the hypercube $Q_n$, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of $M{Q}_n$, and shows that if $M{Q}_n$ ($n\ge 5$) contains at most $n-2$ faulty vertices and/or edges then, for any fault-free edge $uv$ in $M{Q}_n^i(i=0,1)$ and any integer $\ell$ with $7-i⩽\ell ⩽2^n–f_v$, there is a fault-free cycle of length $\ell$ containing the edge $uv$, where $f_v$ is the number of faulty vertices. The result is optimal in some senses.

In a recent paper Cameron, Lakshmanan and Ajith [6] began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable hypergraphs encoding the generating properties of a finite group. In particular, answering a question asked in their paper, we classified the finite solvable groups whose generating hypergraph is the basis hypergraph of a matroid.

Let $G$ be a graph, the zero forcing number $Z(G)$ is the minimum of $|Z|$ over all zero forcing sets $Z\subseteq V(G)$. In this paper, we are interested in studying the zero forcing number of quartic circulant graphs $C_p(s,t)$, where $p$ is an odd prime. Based on the fact that $C_p(s,t)\cong C_p(1,q)$, we give the exact values of the zero forcing number of some specific quartic circulant graphs.

Behera and Panda defined a balancing number as a number b for which the sum of the numbers from $1$ to $b–1$ is equal to the sum of the numbers from $b+1$ to $b+r$ for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.

The secure edge dominating set of a graph $G$ is an edge dominating set $F$ with the property that for each edge $e\in E-F$, there exists $f\in F$ adjacent to $e$ such that $(F-\{f\})\cup \{e\}$ is an edge dominating set. In this paper, we obtained upper bounds for edge domination and secure edge domination number for Mycielski of a tree.

In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected ($HC$) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.

The geometrical properties of a plane determine the tilings that can be built on it. Because of the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries in patterns built on such a surface, which implies the existence of an infinitude of possible tiling families. Using generating functions, we count the vertices of a uniform tiling from any fixed vertex. We count vertices for all families of valence $5$ and for general vertices with valence $6$, with even-sized faces. We also give some general results about the behavior of the vertices and edges of the tilings under consideration.

This study extends the concept of competition graphs to cubic fuzzy competition graphs by introducing additional variations including cubic fuzzy out-neighbourhoods, cubic fuzzy in-neighbourhoods, open neighbourhood cubic fuzzy graphs, closed neighbourhood cubic fuzzy graphs, cubic fuzzy (k) neighbourhood graphs and cubic fuzzy [k]-neighbourhood graphs. These variations provide further insights into the relationships and competition within the graph structure, each with its own defined characteristics and examples. These cubic fuzzy CMGs are further classified as cubic fuzzy k-competition graphs that show competition in the $k$th order between components, $p$-competition cubic fuzzy graphs that concentrate on competition in terms of membership degrees, and $m$-step cubic fuzzy competition graphs that analyze competition in terms of steps. Further, some related results about independent strong vertices and edges have been obtained for these cubic fuzzy competition graph classes. Finally, the proposed concept of cubic fuzzy competition graphs is supported by a numerical example. This example showcases how the framework of cubic fuzzy competition graphs can be practically applied to the predator-prey model to illustrate the representation and analysis of ambiguous information within the graph structures.

A graph $X$ is $k$-spanning cyclable if for any subset $S$ of $k$ distinct vertices there is a 2-factor of $X$ consisting of $k$ cycles such that each vertex in $S$ belongs to a distinct cycle. In this paper, we examine the $k$-spanning cyclability of 4-valent Cayley graphs on Abelian groups.

A path $x_1,x_2,\dots ,x_n$ in a connected graph $G$ that has no edge $x_i{x}_j$ $(j\ge i+3)$ is called a monophonic-triangular path or mt-path. A non-empty subset $M$ of $V(G)$ is a monophonic-triangular set or mt-set of $G$ if every member in $V(G)$ exists in a mt-path joining some pair of members in $M$. The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of $G$ and it is symbolized by $mt(G)$. The general properties satisfied by mt-sets are discussed. Also, we establish $mt$-number boundaries and discover similar results for a few common graphs. Graphs $G$ of order $p$ with $mt(G)=p$, $p–1$, or $p–2$ are characterized.

This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of $V$ and $W$. These values are crucial in determining the Hamming distance and MDS codes in the family of certain constacyclic codes over ${\mathbb{F}}_{p^m}[u]/⟨u^3⟩$, which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.

For a graph $G$ and for non-negative integers $p,q$ and $r$, the triplet $(p,q,r)$ is said to be an admissible triplet, if $3p+4q+6r=|E(G)|$. If $G$ admits a decomposition into $p$ cycles of length $3$, $q$ cycles of length $4$, and $r$ cycles of length $6$ for every admissible triplet $(p,q,r)$, then we say that $G$ has a $\{C_3^p,C_4^q,C_6^r\}$-decomposition. In this paper, the necessary conditions for the existence of $\{C_3^p,C_4^q,C_6^r\}$-decomposition of $K_{\ell ,m,n}(\ell \le m\le n)$ are proved to be sufficient. This affirmatively answers the problem raised in \emph{Decomposing complete tripartite graphs into cycles of lengths $3$ and $4$, Discrete Math. 197/198 (1999), 123-135}. As a corollary, we deduce the main results of \emph{Decomposing complete tripartite graphs into cycles of lengths $3$ and $4$, Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths $3$ and $6$, Austral. J. Combin., 73(1), 220-241 (2019)}.

For a graph $G$ and a subgraph $H$ of a graph $G$, an $H$-decomposition of the graph $G$ is a partition of the edge set of $G$ into subsets $E_i$, $1\le i\le k$, such that each $E_i$ induces a graph isomorphic to $H$. In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the $\lambda$-fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

We classify the geometric hyperplanes of the Segre geometries, that is, direct products of two projective spaces. In order to do so, we use the concept of a generalised duality. We apply the classification to Segre varieties and determine precisely which geometric hyperplanes are induced by hyperplanes of the ambient projective space. As a consequence we find that all geometric hyperplanes are induced by hyperplanes of the ambient projective space if, and only if, the underlying field has order $2$ or $3$.

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, $h^{\ast }(G)$, of a graph $G$, is the least non-negative integer such that from all distributions of $h^{\ast }(G)$ 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 ${\mathrm{s}\mathrm{t}}_{{\gamma }_{oidR}}(G)$ (${\mathrm{s}\mathrm{t}}_{{\gamma }_{oidR}}^{-}(G)$, ${\mathrm{s}\mathrm{t}}_{{\gamma }_{oidR}}^{+}(G)$), 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 ${\mathrm{s}\mathrm{t}}_{{\gamma }_{oidR}}(G)$. In addition, for a tree $T$ with maximum degree $\mathrm{\Delta }$, we show that ${\mathrm{s}\mathrm{t}}_{{\gamma }_{oidR}}(T)=1$ and ${\mathrm{s}\mathrm{t}}_{{\gamma }_{oidR}}^{-}(T)\le \mathrm{\Delta }$, 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 $\epsilon >0$ such that for every caterpillar on $n$ vertices, the value is at most $(1–\epsilon )n+1$.

Let $G$ be a group, and let $c\in {\mathbb{Z}}^{+}\cup \{\mathrm{\infty }\}$. 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 $⟨x_1,x_2⟩$ is not $c$-nilpotent; similarly we let ${\mathrm{\Sigma }}_c(G)$ 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 ${\mathrm{\Sigma }}_c(D_{2k})$, and we show that these two numbers coincide for any given $c$ and $k$.

Let $p>2$ be prime and $r\in \{1,2,\dots ,p-1\}$. 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 $p$, $c_p(pn+r)\equiv 0(\mathrm{mod}2)$ 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 $4\le l\le 2^n$. We consider the following problem of generalizing this property: For a given integer $k$ with $3\le k\le n$, 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 $52\le l\le 2^n$. 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.