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.

A Steiner system \(S(2, k, v)\) is a collection of \(k\)-subsets (blocks) of a \(k\)-set \(V\) such that each \(2\)-subset of \(V\) is contained in exactly one block. We find re-currence relations for \(S(2, k, v)\).

Denote by \(\mathcal{P}(n_1, n_2, n_3)\) the set of all polyphenyl spiders with three legs of lengths \(n_1\), \(n_2\), and \(n_3\). Let \(S^j(n_1, n_2, n_3) \in \mathcal{P}(n_1, n_2, n_3)\) (\(j \in \{1, 2, 3\}\)) be three non-isomorphic polyphenyl spiders with three legs of lengths \(n_1\), \(n_2\), and \(n_3\), and let \(m_k(G)\) and \(i_k(G)\) be the numbers of \(k\)-matchings and \(k\)-independent sets of a graph \(G\), respectively. In this paper, we show that for any \(S^j(n_1, n_2, n_3) \in \mathcal{P}(n_1, n_2, n_3)\) (\(j \in \{1, 2, 3\}\)), we have \(m_k(S_M^3(n_1, n_2, n_3)) \leq m_k(S^j(n_1, n_2, n_3)) \leq m_k(S^j(n_1, n_2, n_3))\) and \(i_k(S_O^1(n_1, n_2, n_3)) \leq i_k(S^j(n_1, n_2, n_3)) \leq i_k(S^3_M(n_1, n_2, n_3))\), with equalities if and only if \(S^j(n_1, n_2, n_3) = S_M^3(n_1, n_2, n_3)\) or \(S^j(n_1, n_2, n_3) = S_O^1(n_1, n_2, n_3)\), where \(S_O^1(n_1, n_2, n_3)\) and \(S_M^3(n_1, n_2, n_3)\) are respectively an ortho-polyphenyl spider and a meta-polyphenyl spider.