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.

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.

The hyper Wiener index of a connected graph \(G\) is defined as
\(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2}\sum_{(u,v) \in V(G)} d(u,v)^2\) where \(d(u, v)\) is the distance between vertices \(u,v \in V(G)\).
In this paper we find an exact expression for hyper Wiener index of \(HC_6[p, q]\), the zigzag polyhex nanotori.

In this paper, we classify all optimal linear \([n, n/2]\) codes over \(\mathbb{Z}_4\) up to length \(n = 8\), and determine the number of optimal codes which are self-dual and formally self-dual. Optimal codes with linear binary images are identified. In particular, we show that for length \(8\), there are nine optimal codes for the Hamming distance, one optimal code for the Lee distance, and two optimal codes for the Euclidean distance.

In this paper, we show that if \(k \geq \frac{v+2}{4}\), where \(v\) denotes the order of a graph, a non-bipartite graph \(G\) is \(k\)-extendable if and only if it is \(2k\)-factor-critical. If \(k \geq \frac{v-3}{4}\), a graph \(G\) is \(k\)-extendable if and only if it is \((2k+1)\)-factor-critical. We also give examples to show that the two bounds are best possible. Our results are answers to a problem posted by Favaron \([3]\) and Yu \([11]\).

The edge-neighbor-scattering number of a graph \(G\) is defined to be \(EN_S(G) = \max\limits_{S\subseteq E(G)}\{w(G/S) -\mid |S|\}\) where \(S\) is any edge-cut-strategy of \(G\), \(w(G/S)\) is the number of the components of \(G/S\). In this paper, we give edge-neighbor-scattering number of some special classes of graphs, and then mainly discuss the general properties of the parameter.

Let \(F(x,y) = ax^2 + bxy + cy^2\) be a binary quadratic form of discriminant \(\Delta = b^2 – 4ac\) for \(a,b,c \in \mathbb{Z}\), let \(p\) be a prime number and let \(\mathbb{F}_p\) be a finite field. In this paper we formulate the number of integer solutions of cubic congruence \(x^3 + ax^2 + bx + c \equiv 0 \pmod{p}\) over \(\mathbb{F}_p\), for two specific binary quadratic forms \(F_1^k(x,y) = x^2 + kxy + ky^2\) and \(F_2^k(x,y) = kx^2 + kxy + k^2y^2\) for integer \(k\) such that \(1 \leq k \leq 9\). Later we consider representation of primes by \(F_1^k\) and \(F_2^k\).

A subset \(S \subseteq V(G)\) is independent if no two vertices of \(S\) are adjacent in \(G\). In this paper we study the number of independent sets which meets the set of leaves in a tree. In particular we determine the smallest number and the largest number of these sets among \(n\)-vertex trees. In each case we characterize the extremal graphs.

A graph \(G\) is called super edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\). Yasuhiro Fukuchi proved that the generalized Petersen graph \(P(n, 2)\) is super edge-magic for odd \(n \geq 3\). In this paper, we show that the generalized Petersen graph \(P(n,3)\) is super edge-magic for odd \(n \geq 5\).

For any integer \(k\), two tournaments \(T\) and \(T’\), on the same finite set \(V\) are \(k\)-similar, whenever they have the same score vector, and for every tournament \(H\) of size \(k\) the number of subtournaments of \(T\) (resp. \(T’\)) isomorphic to \(H\) is the same. We study the \(4\)-similarity. According to the decomposability, we construct three infinite classes of pairs of non-isomorphic \(4\)-similar tournaments.

In this paper, we define the Pell and Pell-Lucas \(p\)-numbers and derive the analytical formulas for these numbers. These formulas are similar to Binet’s formula for the classical Pell numbers.

A graph \(G\) is called resonant if the boundary of each face of \(G\) is an \(F\)-alternating closed trail with respect to some \(f\)-factor \(F\) of \(G\). We show that a plane bipartite graph \(G\) is resonant if and only if it is connected and each edge of \(G\) is contained in an \(f\)-factor and not in another \(f\)-factor.

Let \(P_k\) denote a path with \(k\) vertices and \(k-1\) edges. And let \(\lambda K_{n,n}\) denote the \(\lambda\)-fold complete bipartite graph with both parts of size \(n\). A \(P_k\)-decomposition \(\mathcal{D}\) of \(\lambda K_{n,n}\) is a family of subgraphs of \(\lambda K_{n,n}\) whose edge sets form a partition of the edge set of \(\lambda K_{n,n}\), such that each member of \(\mathcal{G}\) is isomorphic to \(P_k\). Necessary conditions for the existence of a \(P_k\)-decomposition of \(\lambda K_{n,n}\) are (i) \(\lambda n^2 \equiv 0 \pmod{k-1}\) and (ii) \(k \leq n+1\) if \(\lambda=1\) and \(n\) is odd, or \(k \leq 2n\) if \(\lambda \geq 2\) or \(n\) is even. In this paper, we show these necessary conditions are sufficient except for the possibility of the case that \(k=3\), \(n=15\), and \(k=28\).

We describe a technique for producing self-dual codes over rings and fields from symmetric designs. We give special attention to biplanes and determine the minimum weights of the codes formed from these designs. We give numerous examples of self-dual codes constructed including an optimal code of length \(22\) over \(\mathbb{Z}_4\) with respect to the Hamming metric from the biplane of order \(3\).

The distance graph \(G(S, D)\) has vertex set \(V(G(S, D)) = S \cup \mathbb{R}^n\) and two vertices \(u\) and \(v\) are adjacent if and only if their distance \(d(u, v)\) is an element of the distance set \(D \subseteq \mathbb{R}_+\).

We determine the chromatic index, the choice index, the total chromatic number and the total choice number of all distance graphs \(G(\mathbb{R}, D)\), \(G(\mathbb{Q}, D)\) and \(G(\mathbb{Z}, D)\) transferring a theorem of de Bruijn and Erdős on infinite graphs. Moreover, we prove that \(|D| + 1\) is an upper bound for the chromatic number and the choice number of \(G(S,D)\), \(S \subseteq \mathbb{R}\).

Some results on combinatorial aspects of block designs using the complementary property have been obtained. The results pertain to non-existence of partially balanced incomplete block (PBIB) designs and identification of new \(2\)-associate and \(3\)-associate PBIB designs. A method of construction of extended group divisible (EGD) designs with three factors using self-complementary rectangular designs has also been given. Some rectangular designs have also been obtained using self-complementary balanced incomplete block designs. Catalogues of EGD designs and rectangular designs obtainable from these methods of construction, with number of replications \(\leq 10\) and block size \(\leq 10\) have been prepared.

For any simple graph \(H\), let \(\sigma(H, n)\) be the minimum \(m\) so that for any realizable degree sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with sum of degrees at least \(m\), there exists an \(n\)-vertex graph \(G\) witnessing \(\pi\) that contains \(H\) as a weak subgraph. Let \(F_{k}\) denote the friendship graph on \(2k+1\) vertices, that is, the graph of \(k\) triangles intersecting in a single vertex. In this paper, for \(n\) sufficiently large, \(\sigma(F_{k},n)\) is determined precisely.

Let \(C\) be a plane convex body, and let \(l(ab)\) be the Euclidean length of a longest chord of \(C\) parallel to the segment \(ab\) in \(C\). By the relative length of \(ab\) in a convex body \(C\), we mean the ratio of the Euclidean length of \(ab\) to \(\frac{l(ab)}{2}\). We say that a side \(ab\) of a convex \(n\)-gon is relatively short if the relative length of \(ab\) is not greater than the relative length of a side of the regular \(n\)-gon. In this article, we provide a significant sufficient condition for a convex hexagon to have a relatively short side.

This paper studies families of self-orthogonal codes over \(\mathbb{Z}_4\). We show that the simplex codes (of Type \(a\) and Type \(\beta\)) are self-orthogonal. We answer the question of \(\mathbb{Z}_4\)-linearity for some codes obtained from projective planes of even order. A new family of self-orthogonal codes over \(\mathbb{Z}_4\) is constructed via projective planes of odd order. Properties such as self-orthogonality, weight distribution, etc. are studied. Finally, some self-orthogonal codes constructed from twistulant matrices are presented.

A complete paired comparison digraph \(D\) is a directed graph in which \(xy\) is an arc for all vertices \(x,y\) in \(D\), and to each arc we assign a real number \(0 \leq a \leq 1\) called a weight such that if \(xy\) has weight \(a\) then \(yx\) has weight \(1 – a\). We say that two vertices \(x, y\) dominate a third \(z\) if the weights on \(xz\) and \(yz\) sum to at least \(1\). If \(x\) and \(y\) dominate all other vertices in a complete paired comparison digraph, then we say they are a dominant pair. We construct the domination graph of a complete paired comparison digraph \(D\) on the same vertices as \(D\) with an edge between \(x\) and \(y\) if \(x\) and \(y\) form a dominant pair in \(D\). In this paper, we characterize connected domination graphs of complete paired comparison digraphs. We also characterize the domination graphs of complete paired comparison digraphs with no arc weight of \(.5\).

A graph \(G\) is a \((d,d+k)\)-graph, if the degree of each vertex of \(G\) is between \(d\) and \(d+k\). Let \(p > 0\) and \(d+k \geq 2\) be integers. If \(G\) is a \((d,d+k)\)-graph of order \(n\) with at most \(p\) odd components and without a matching \(M\) of size \(2|M| = n – p\), then we show in this paper that

1. \(n \geq 2d+p+2\) when \(p \leq k-2\),
2. \(n \geq 2\left\lceil \frac{d(p+2)}{k} \right\rceil +p +2\) when \(p \geq k-1\).

Corresponding results for \(0 \leq p \leq 1\) and \(0 \leq k \leq 1\) were given by Wallis \([6]\), Zhao \([8]\), and Volkmann \([5]\).

Examples will show that the given bounds (i) and (ii) are best possible.

In this paper, we prove that the cycle \(C_n\) with parallel chords and the cycle \(C_n\) with parallel \(P_k\)-chords are cordial for any odd positive integer \(k \geq 3\) and for all \(n \geq 4\) except for \(n \equiv 4r + 2, r \geq 1\). Further, we show that every even-multiple subdivision of any graph \(G\) is cordial and we show that every graph is a subgraph of a cordial graph.

A hypergraph is linear if no two distinct edges intersect in more than one vertex. A long standing conjecture of Erdős, Faber, and Lovász states that if a linear hypergraph has \(n\) edges, each of size \(n\), then its vertices can be properly colored with \(n\) colors. We prove the correctness of the conjecture for a new, infinite class of linear hypergraphs.

We use a computer to show that the crossing number of generalized Petersen graph \(P(10,3)\) is six.

Let \(G\) be a graph in which each vertex has been coloured using one of \(k\) colours, say \(c_1,c_2,\ldots,c_k\) If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices coloured \(c_i\), \(i = 1,2,\ldots,k\), and \(|n_i – n_j| \leq 1\) for any \(i,j \in \{1,2,\ldots,k\}\), then \(C\) is equitably \(k\)-coloured. An \(m\)-cycle decomposition \(C\) of a graph \(G\) is equitably \(k\)-colourable if the vertices of \(G\) can be coloured so that every \(m\)-cycle in \(C\) is equitably \(k\)-coloured. For \(m = 4,5\) and \(6\), we completely settle the existence problem for equitably \(2\)-colourable \(m\)-cycle decompositions of complete graphs and complete graphs with the edges of a \(1\)-factor removed.