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.

Two classes of regular Cayley maps, balanced and antibalanced, have long been understood, see \([12,11]\). A recent generalization is that of an e-balanced map, see \([7,2,5,8]\). These maps can be described using the power function introduced in \([4]\); e-balanced maps are the ones with constant power functions on the generating set. In this paper we examine a further generalization to the situation where the power function alternates between two values.

In this paper, we obtain the spectral norm and eigenvalues of circulant matrices with Horadam’s numbers. Furthermore, we define the semicirculant matrix with these numbers and give the Euclidean norm of this matrix.

We denote by \(G(n)\) the graph obtained by removing a Hamilton cycle from the complete graph \(K_n\). In this paper, we calculate the lower bound for the minimum number of monochromatic triangles in any \(2\)-edge coloring of \(G(n)\) using the weight method. Also, by explicit constructions, we give an upper bound for the minimum number of monochromatic triangles in \(2\)-edge coloring of \(G(n)\) and the difference between our lower and upper bound is just two.

In this paper, it is proved that the \(h\)-chromatic uniqueness of the linear \(h\)-hypergraph consisting of two cycles of lengths \(p\) and \(q\) having \(r\) edges in common when \(p=q\), \(2 \leq r \leq p-2\), and \(h \geq 3\). We also obtain the chromatic polynomial of a connected unicyclic linear \(h\)-hypergraph and show that every \(h\)-uniform cycle of length three is not chromatically unique for \(h \geq 3\).

The projection of binary linear block codes of length \(4m\) on \(\mathbb{F}_4^m\) is considered. Three types of projections, namely projections \(SE\), \(E\), and \(O\) are introduced. The BCH codes, Golay codes, Reed-Muller codes, and the quadratic residue code \(q_{32}\) are examined.