Let \(\vec{P_l}\) be the directed path on \(r\) vertices and \(\lambda K^*_{m,n}\) be the symmetric complete bipartite multi-digraph with two partite sets having \(m\) and \(n\) vertices. A \(\vec{P_l}\)-factorization of \(\lambda K^*_{m,n}\) is a set of arc-disjoint \(\vec{P_l}\)-factors of \(\lambda K^*_{m,n}\), which is a partition of the set of arcs of \(\lambda K^*_{m,n}\). In this paper, it is shown that a necessary and sufficient condition for the existence of a \(\vec{P}_{2k+l}\)-factorization of \(\lambda K^*_{m,n}\) for any positive integer \(k\).

Let \(G = (V, E)\) be a finite non-empty graph. A vertex-magic total labeling (VMTL) is a bijection \(\lambda\) from \(V \cup E\) to the set of consecutive integers \(\{1, 2, \ldots, |V| + |E|\}\) with the property that for every \(v \in V\), \(\lambda(v) + \sum_{w \in N(v)} \lambda(vw) = h\), for some constant \(h\). Such a labeling is called super if the vertex labels are \(1, 2, \ldots, |V|\).
There are some results known about super VMTLs of \(kG\) only when the graph \(G\) has a super VMTL. In this paper, we focus on the case when \(G\) is the complete graph \(K_n\). It was shown that a super VMTL of \(kK_n\) exists for \(n\) odd and any \(k\), for \(4 < n \equiv 0 \pmod{4}\) and any \(k\), and for \(n = 4\) and \(k\) even. We continue the study and examine the graph \(kK_n\) for \(n \equiv 2 \pmod{4}\). Let \(n = 4l + 2\) for a positive integer \(l\). The graph \(kK_{4l+2}\) does not admit a super VMTL for \(k\) odd. We give a large number of super VMTLs of \(kK_{4l+2}\) for any even \(k\) based on super VMTLs of \(4K_{2l+1}\).

For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph. Let \(K_m – H\) be the graph obtained from \(K_m\) by removing the edge set \(E(H)\), where \(H\) is a subgraph of \(K_m\). In this paper, we characterize the potentially \(K_6 – C_4\)-graphic sequences. This characterization implies a theorem due to Hu and Lai \([7]\).

Double Fibonacci sequences \((x_{n,k})\) are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.

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 \(m \geq 3\).