A set \(S \subseteq V\) is a dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is either in \(S\) or is adjacent to a vertex in \(S\). A vertex is said to dominate itself and all its neighbors. The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). In terms of a chess board problem, let \(X_n\) be the graph for chess piece \(X\) on the square of side \(n\). Thus, \(\gamma(X_n)\) is the domination number for chess piece \(X\) on the square of side \(n\). In 1964, Yaglom and Yaglom established that \(\gamma(K_n) = \left\lceil \frac{n+2}{2} \right\rceil^2\). This extends to \(\gamma(K_{m,n}) = \left\lceil \frac{m+2}{3} \right\rceil \left\lceil \frac{n+2}{3} \right\rceil\) for the rectangular board. A set \(S \subseteq V\) is a total dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is adjacent to a vertex in \(S\). A vertex is said to dominate its neighbors but not itself. The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). In 1995, Garnick and Nieuwejaar conducted an analysis of the total domination numbers for the king’s graph on the \(m \times n\) board. In this paper, we note an error in one portion of their analysis and provide a correct general upper bound for \(\gamma_t(K_{m,n})\). Furthermore, we state improved upper bounds for \(\gamma_t(K_n)\).

A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually the positive integers). An \((a, d)\)-edge-antimagic total labeling of a graph with \(p\) vertices and \(q\) edges is a one-to-one mapping that takes the vertices and edges onto the integers \(1, 2, \ldots, p + q\), such that the sums of the label on the edges and the labels of their end points form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called \({super}\) if the smallest possible labels appear on the vertices. In this paper, we study the super \((a, 2)\)-edge-antimagic total labelings of disconnected graphs. We also present some necessary conditions for the existence of \((a, d)\)-edge-antimagic total labelings for \(d\) even.

Fault tolerance is an important property of network performance. A graph \(G\) is \(k\)-edge-fault conditional Hamiltonian if \(G – F\) is Hamiltonian for every \(F \subset E(G)\) with \(|F| \leq k\) and \(\delta(G – F) \geq 2\). In this paper, we show that for \(n \geq 4\), the \(n\)-dimensional star graph \(S_n\) is \((3n – 10)\)-edge-fault conditional Hamiltonian.

In this paper, we characterize all spacelike, timelike, and null curves lying on the pseudohyperbolic space \({H}^{4}_{v-1}\), in Minkowski space \({E}^5_v\). Moreover, we prove that there are no timelike and no null curves lying on the pseudohyperbolic space \({H}^{4}_{v-1}\) in \({E}^5_v\).

The local-restricted-edge-connectivity \(\lambda'(e, f)\) of two nonadjacent edges \(e\) and \(f\) in a graph \(G\) is the maximum number of edge-disjoint \(e\)-\(f\) paths in \(G\). It is clear that \(\lambda'(G) = \min\{\lambda'(e, f) \mid e \text{ and } f \text{ are nonadjacent edges in } G\}\), and \(\lambda'(e, f) \leq \min\{\xi(e), \xi(f)\}\) for all pairs \(e\) and \(f\) of nonadjacent edges in \(G\), where \(\lambda(G)\), \(\xi(e)\), and \(\xi(f)\) denote the restricted-edge-connectivity of \(G\), the edge-degree of edges \(e\) and \(f\), respectively. Let \(\xi(G)\) be the minimum edge-degree of \(G\). We call a graph \(G\) optimally restricted-edge-connected when \(\lambda'(G) = \xi(G)\) and optimally local-restricted-edge-connected if \(\lambda'(e, f) = \min\{\xi(e),\xi(f)\}\) for all pairs \(e\) and \(f\) of nonadjacent edges in \(G\). In this paper, we show that some known sufficient conditions that guarantee that a graph is optimally restricted-edge-connected also guarantee that it is optimally local-restricted-edge-connected.

In 1982, Beutelspacher and Brestovansky proved that for every integer \(m \geq 3\), the \(2\)-color Rado number of the equation
\[x_1+x_2+ \ldots + x_{m-1}=x_m\]
is \(m^2 – m – 1\). In 2008, Schaal and Vestal proved that, for every \(m \geq 6\), the \(2\)-color Rado number of
\[x_1+x_2+ \ldots + x_{m-1}=2x_m\]
is \(\left\lceil \frac{m-1}{2}\left\lceil \frac{m-1}{2} \right\rceil \right\rceil \). Here, we prove that, for every integer \(a \geq 3\) and every \(m \geq 2a^2 – a + 2\), the 2-color Rado number of
\[x_1+x_2+ \ldots + x_{m-1}=ax_m\]
is \(\left\lceil \frac{m-1}{a}\left\lceil \frac{m-1}{a} \right\rceil \right\rceil\). For the case \(a = 3\), we show that our formula gives the Rado number for all \(m \geq 7\), and we determine the Rado number for all \(m \geq 3\).

The general Randic index \(R_{-\alpha}(G)\) of a graph \(G\), defined by a real number \(\alpha\), is the sum of \((d(u)d(v))^{-\alpha}\) over all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we have discussed some properties of the Max Tree which has the maximum general Randic index \(R_{-\alpha}(G)\), where \(\alpha \in (\alpha_0,2)\). Based on these properties, we are able to obtain the structure of the Max Tree among all trees of order \(k \geq 3\). Thus, the maximal value of \(R_{-\alpha}(G)\) follows easily.

A \(\lambda\)-fold \(G\)-design of order \(n\) is a pair \((X, {B})\), where \(X\) is a set of \(n\) vertices and \({B}\) is a collection of edge-disjoint copies of the simple graph \(G\), called blocks, which partitions the edge set of \(K_n\) (the undirected complete graph with \(n\) vertices) with vertex set \(X\). Let \((X, {B})\) be a \(G\)-design and \(H\) be a subgraph of \(G\). For each block \(B \in \mathcal{B}\), partition \(B\) into copies of \(H\) and \(G \setminus H\) and place the copy of \(H\) in \({B}(H)\) and the edges belonging to the copy of \(G \setminus H\) in \({D}(G \setminus H)\). Now, if the edges belonging to \({D}(G \setminus H)\) can be arranged into a collection \({D}_H\) of copies of \(H\), then \((X, {B}(H) \cup {D}(H))\) is a \(\lambda\)-fold \(H\)-design of order \(n\) and is called a metamorphosis of the \(\lambda\)-fold \(G\)-design \((X, {B})\) into a \(\lambda\)-fold \(H\)-design, denoted by \((G > H) – M_\lambda(n)\).
In this paper, the existence of a \((G > H) – M_\lambda(n)\) for graph designs will be presented, variations of this problem will be explained, and recent developments will be surveyed.

For an integer \(k \geq 1\) and a graph \(G = (V, E)\), a subset \(S\) of the vertex set \(V\) is \(k\)-independent in \(G\) if the maximum degree of the subgraph induced by the vertices of \(S\) is less than or equal to \(k – 1\). The \(k\)-independence number \(\beta_k(G)\) of \(G\) is the maximum cardinality of a \(k\)-independent set of \(G\). A set \(S\) of \(V\) is \(k\)-Co-independent in \(G\) if \(S\) is \(k\)-independent in the complement of \(G\). The \(k\)-Co-independence number \(\omega_k(G)\) of \(G\) is the maximum size of a \(k\)-Co-independent set in \(G\). The sequences \((\beta_k)\) and \((\omega_k)\) are weakly increasing. We define the \(k\)-chromatic number or \(k\)-independence partition number \(\chi_k(G)\) of \(G\) as the smallest integer \(m\) such that \(G\) admits a partition of its vertices into \(m\) \(k\)-independent sets and the \(k\)-Co-independence partition number \(\theta_k(G)\) of \(G\) as the smallest integer \(m\) such that \(G\) admits a partition of its vertices into \(m\) \(k\)-Co-independent sets. The sequences \((\chi_k)\) and \((\theta_k)\) are weakly decreasing. In this paper, we mainly present bounds on these four parameters, some of which are extensions of well-known classical results.

It is proved that if \(G\) is a plane embedding of a \(K_4\)-minor-free graph, then \(G\) is coupled \(5\)-choosable; that is, if every vertex and every face of \(G\) is given a list of \(5\) colours, then each of these ele-ments can be given a colour from its list such that no two adjacent or incident elements are given the same colour. Using this result it is proved also that if \(G\) is a plane embedding of a \(K_{2,3}\),\(3\)-minor-free graph or a \((\bar{K}_2 + (K_1 \cup K_2))\)-minor-free graph, then \(G\) is coupled \(5\)-choosable. All results here are sharp, even for outerplane graphs.