In this paper, we show that for every modular lattice \(L\), if its size is at least three times its excess, then each component of its direct product decomposition is isomorphic to one of the following: a Boolean lattice of rank one \(B_1\), a chain of length two \(3\), a diamond \(M_3\), and \(M_4\), where \(M_n\) is a modular lattice of rank two which has exactly \(n\) atoms.

Using algebraic curves, it will be proven that large partial unitals can be embedded into unitals and large \((k,n)\)-arcs into maximal arcs.

In a set equipped with a binary operation, \((S, \cdot)\), a subset \(U\) is defined to be avoidable if there exists a partition \(\{A, B\}\) of \(S\) such that no element of \(U\) is the product of two distinct elements of \(A\) or of two distinct elements of \(B\). For more than two decades, avoidable sets in the natural numbers (under addition) have been studied by renowned mathematicians such as Erdős, and a few families of sets have been shown to be avoidable in that setting. In this paper, we investigate the generalized notion of an avoidable set and determine the avoidable sets in several families of groups; previous work in this field considered only the case \((S, \cdot) = (\mathbb{N}, +)\).

This paper studied the problems of counting independent sets, maximal independent sets, and maximum independent sets of a graph from an algorithmic point of view. In particular, we present linear-time algorithms for these problems in trees and unicyclic graphs.

The Stirling numbers of first kind and Stirling numbers of second kind, denoted by \(s(n,k)\) and \(S(n,k)\) respectively, arise in a variety of combinatorial contexts. There are several algebraic and combinatorial relationships between them. Here, we state and prove four new identities concerning the determinants of matrices whose entries are unsigned Stirling numbers of first kind and Stirling numbers of second kind. We also observe an interrelationship between them based on our identities.

We generalize a construction by Treash of a Steiner triple system on \(2v+1\) points that embeds a Steiner triple system on \(v\) points. We show that any Steiner quadruple system on \(v+1\) points may be embedded in a Steiner quadruple system on \(2v+2\) points.

A \((\lambda K_n, G)\)-design is a partition of the edges of \(\lambda K_n\), into sub-graphs each of which is isomorphic to \(G\). In this paper, we investigate the existence of \((K_n, G_{16})\)-design and \((K_n, G_{20})\)-design, and prove that the necessary conditions for the existence of the two classes of graph designs are also sufficient.

Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \(ae\) is the absolute value of the difference of the labels of \(a\) and \(e\). A labeling of the vertices of a graph of order \(p\) is minimally \(k\)-equitable if the vertices are labeled with elements of \({1,2, \ldots, p}\) and in the induced labeling of its edges, every label either occurs exactly \(k\) times or does not occur at all. We prove that the corona graph \(C_{2n}OK_1\) is minimally \(4\)-equitable.

A set of Bishops cover a board if they attack all unoccupied squares. What is the minimum number of Bishops needed to cover an \(k \times n\) board \(?\) Yaglom and Yaglom showed that if \(k = n\), the answer is \(n\). We extend this result by showing that the minimum is \(2\lfloor \frac{n}{2}\rfloor\) if \(k 2k > 2\), a cover is given with \(2\lfloor\frac{k+n}{2}\rfloor\) Bishops. We conjecture that this is the minimum value. This conjecture is verified when \(k \leq 3\) or \(n \leq 2k + 5\).

It is proved that the following graphs are harmonious:(1) shell graphs (2) cycles with the maximum possible number of concurrent alternate chords (3) Some families of multiple shells

In this paper, we determine all harmonious graphs of order \(6\).
All graphs in this paper are finite, simple and undirected. We shall use the basic notation and terminology of graph theory as in [1].

Let \(R(n)\) denote the number of two-color partitions of \(n\). We obtain several identities concerning \(R(n)\).

We show that if \(M(n, m)\) denotes the time of a \((u, v)\)-minimum cut computation in a directed graph with \(n \geq 2\) nodes, \(m\) edges, and \(s\) and \(t\) are two distinct given nodes, then there exists an algorithm with \(O(n^2m+n\cdot M(n, m))\) running time for the directed minimum odd (or even) \((s, t)\)-cut problem and for its certain generalizations.

Basic properties of in-degree distribution of a general model of random digraphs \(D(n, \mathcal{P})\) are presented. Then some relations between random digraphs \(D(n, \mathcal{P})\) for different probability distributions \(\mathcal{P}\)’s are studied. In this context, a problem of the existence of a threshold function for every monotone digraph property of \(D(n, \mathcal{P})\) is discussed.

For a given structure (graph, multigraph, or pseudograph) \(G\) and an integer \(r \geq \Delta(G)\), a smallest inducing \(r\)-regularization of \(G\) (which is an \(r\)-regular superstructure of the smallest possible order, with bounded edge multiplicities, and containing \(G\) as an induced substructure) is constructed.

It is an established fact that some graph-theoretic extremal questions play an important part in the investigation of communication network vulnerability. Questions concerning the realizability of graph invariants are generalizations of these extremal problems. We define a \((p, q, \lambda, \delta)\) graph as a graph having \(p\) points, \(q\) lines, line connectivity \(\lambda\) and minimum degree \(\delta\). An arbitrary quadruple of integers \((a, b, c, d)\) is called \((p, q, \lambda, \delta)\) realizable if there is a \((p, q, \lambda, \delta)\) graph with \(p = a, q = b, \lambda = c\), and \(\delta = d\). Inequalities representing necessary and sufficient conditions for a quadruple to be \((p, q, \lambda, \delta)\) realizable are derived. In recent papers, the author gave necessary and sufficient conditions for \((p, q, \kappa, \Delta), (p, q, \lambda, \Delta), (p, q, \delta, \Delta)\) and \((p, q, \kappa, \delta)\) realizability, where \(\Delta\) denotes the maximum degree for all points in a graph and \(\lambda\) denotes the point connectivity of a graph. Boesch and Suffel gave the solutions for \((p, q, \kappa), (p, q, \lambda), (p, q, \delta), (p, \Delta, \delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability in earlier manuscripts.

An aperiodic perfect map (APM) is an array with the property that each possible array of certain size, called a window, arises exactly once as a subarray in the array. In this article, we give some constructions which imply a complete answer for the existence of APMs with \(2 \times 2\) windows for any alphabet size.

A \(4\)-regular graph \(G\) is called a \(4\)-circulant if its adjacency matrix \(A(G)\) is a circulant matrix. Because of the special structure of the eigenvalues of \(A(G)\), the rank of such graphs is completely determined. We show how all disconnected \(4\)-circulants are made up of connected \(4\)-circulants and classify all connected \(4\)-circulants as isomorphic to one of two basic types.

Let \([n, k, d; g]\)-codes be linear codes of length \(n\), dimension \(k\) and minimum Hamming distance \(d\) over \(\mathrm{GF}(g)\). Let \(d_8(n, k)\) be the maximum possible minimum Hamming distance of a linear \([n, k, d; 8]\)-code for given values of \(n\) and \(k\). In this paper, twenty-two new linear codes over \(\mathrm{GF}(8)\) are constructed which improve the bounds on \(d_8(n, k)\).

We find new full orthogonal designs in order \(56\) and show that of
\(1285\) possible \(OD(56; s_1, s_2, s_3,56 – s_1 – s_2 – s_3)\) \(163\) are known, of
\(261\) possible \(OD(56; s_1, s_2, 56 – s_1 – s_2)\) \(179\) are known. All possible
\(OD(56; s_1,56 – s_1)\) are known.

Sattolo has presented an algorithm to generate cyclic permutations at random. In this note, the two parameters “number of moves” and “distance” are analyzed.

In this paper, we shall classify the self-complementary graphs with minimum degree exactly \(2\).

A graphical partition of the even integer \(n\) is a partition of \(n\) where each part of the partition is the degree of a vertex in a simple graph and the degree sum of the graph is \(n\). In this note, we consider the problem of enumerating a subset of these partitions, known as graphical forest partitions, graphical partitions whose parts are the degrees of the vertices of forests (disjoint unions of trees). We shall prove that

\[gf(2k) = p(0) + p(1) + p(2) + \cdots + p(k-1)\]

where \(g_f(2k)\) is the number of graphical forest partitions of \(2k\) and \(p(j)\) is the ordinary partition function which counts the number of integer partitions of \(j\).

We make further progress towards the forbidden-induced-subgraph characterization of the graphs with Hall number \(\leq 2\). We solve several problems posed in [4] and, in the process, describe all “partial wheel” graphs with Hall number \(\geq 2\) with every proper induced subgraph having Hall number \(\leq 2\).

A radio labeling of a connected graph $G$ is an assignment of distinct, positive integers to the vertices of \(G\), with \(x \in V(G)\) labeled \(c(x)\), such that

\[d(u, v) + |c(u) – c(v)| \geq 1 + diam(G)\]

for every two distinct vertices \(u,v\) of \(G\), where \(diam(G)\) is the diameter of \(G\). The radio number \(rn(c)\) of a radio labeling \(c\) of \(G\) is the maximum label assigned to a vertex of \(G\). The radio number \(rn(G)\) of \(G\) is \(\min\{rn(c)\}\) over all radio labelings \(c\) of \(G\). Radio numbers of cycles are discussed and upper and lower bounds are presented.

Dudeney’s round table problem was proposed about one hundred years ago. It is already solved when the number of people is even, but it is still unsettled except for only a few cases when the number of people is odd.

In this paper, a solution of Dudeney’s round table problem is given when \(n = p+2\), where \(p\) is an odd prime number such that \(2\) is the square of a primitive root of \(\mathrm{GF}(p)\), \(p \equiv 1 \pmod{4}\), and \(3\) is not a quadratic residue modulo \(p\).