Minimal blocking sets of class \([h,k]\) with respect to the external lines to an elliptic quadric of \(\text{PG}(3,q)\), \(q \geq 5\) prime, are characterized.

For every integer \(c\) and every positive integer \(k\), let \(n = r(c, k)\) be the least integer, provided that it exists, such that for every coloring

\[\Delta: \{1,2,\ldots,n\} \rightarrow \{0,1\},\]

there exist three integers, \(x_1, x_2, x_3\), (not necessarily distinct) such that
\[\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\]
and
\[x_1+x_2+c= kx_3.\]

If such an integer does not exist, then let \(r(c, k) = \infty\). The main result of this paper is that

\[r(c,2) =
\begin{cases}
|c|+1 & \text{if } c \text{ is even} \\
\infty & \text{if } c \text{ is odd}
\end{cases}\]

for every integer \(c\). In addition, a lower bound is found for \(r(c, k)\) for all integers \(c\) and positive integers \(k\) and linear upper and lower bounds are found for \(r(c, 3)\) for all positive integers \(c\).

Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\) with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0,3 \pmod 4\). The conjecture has been shown true for \(n = 3,5,6,7,4k\). In this paper, the conjecture is shown to be true for \(n = 9\).

In this paper, we define the hyperbolic modified Pell functions by the modified Pell sequence and classical hyperbolic functions. Afterwards, we investigate the properties of the modified Pell functions.

Deza and Grishukhin studied \(3\)-valent maps \(M_n{(p,q)}\) consisting of a ring of \(n\) \(g\)-gons whose inner and outer domains are filled by \(p\)-gons. They described the conditions for \(n, p, q\) under which such a map may exist and presented several infinite families of them. We extend their results by presenting several new maps concerning mainly large values of \(n\) and \(q\).

A simple, undirected \(2\)-connected graph \(G\) of order \(n\) belongs to the class \(\mathcal{B}(n,\theta)\), \(\theta \geq 0\) if \(2(d(x) + d(y) + d(z)) \geq 3(n – 1 – \theta)\) holds for all independent triples \(\{x,y,z\}\) of vertices. It is known (Bondy’s theorem for \(2\)-connected graphs) that \(G\) is hamiltonian if \(\theta = 0\). In this paper we give a full characterization of graphs \(G\) in \(\mathcal{B}(n,\theta)\), \(\theta \leq 2\) in terms of their dual hamiltonian closure.