We present sufficient conditions for the existence of a \(k\)-factor in a simple graph depending on \(\sigma_2(G)\) and the neighbourhood of independent sets in our first theorem and on \(\sigma_2(G)\) and \(\alpha(G)\) in the second one.

It is well known that a necessary condition for the existence of a \((v, 4, 1)\)-RPMD is \(v \equiv 0 \text{ or } 1 \pmod{4}\) and the existence of \((v, 4, 1)\)-RPMDs for \(v \equiv 1 \pmod{4}\) has been completely settled.
In this paper, we shall introduce the concept of \((v, k, 1)\)-nearly-RPMDs and use it to obtain some new construction methods for \((v, k, 1)\)-RPMDs with \(v \equiv 0 \pmod{k}\). As an application, we shall show that a \((v, 4, 1)\)-RPMD exists for all integers \(v \geq 4\) where \(v \equiv 0 \pmod{4}\), except for \(v = 4, 8\) and with at most \(49\) possible exceptions of which the largest is \(336\).
It is also well known that a \((v, k, 1)\)-RPMD exists for all sufficiently large \(v\) with \(k \geq 3\) and \(v \equiv 1 \pmod{k}\), and a \((v, k, 1)\)-PMD exists with \(v(v – 1) \equiv 0 \pmod{k}\) for the case when \(k\) is an odd prime and \(v\) is sufficiently large. In this paper, we shall show that there exists a \((v, k, 1)\)-RPMD for all sufficiently large \(v\) with \(v \equiv 0 \pmod{k}\), and there exists a \((v, k,\lambda)\)-PMD for all sufficiently large \(v\) with \(\lambda v(v – 1) \equiv 0 \pmod{k}\).

In this paper we have investigated harmonious labelings of \(p\)-stars, where a \(p\)-star of length \(x\) is a star tree in which each edge is a path of length \(k\). We have also demonstrated an application of the labelings to \(k\) disjoint \(p\)-cycles.

We show that if a graph \(G\) has \(n\) non-isomorphic \(2\)-vertex deleted subgraphs then \(G\) has at most \(n\) distinct degrees. In addition, we prove that if \(G\) has \(3\) non-isomorphic \(3\)-vertex deleted subgraphs then \(G\) has at most \(3\) different degrees.

Observability of a graph is the least \(k\) admitting a proper coloring of its edges by \(k\) colors in such a way that each vertex is identifiable by the set of colors of its incident edges. It is shown that for \(p \geq 3\) and \(q \geq 2\) the complete \(p\)-partite graph with all parts of cardinality \(q\) has observability \((p-1)q+2\).

Let \(V\) be a finite set of order \(\nu\). A \((\nu,\kappa,\lambda)\) packing design of index \(\lambda\) and block size \(\kappa\) is a collection of \(\kappa\)-element subsets, called blocks, such that every \(2\)-subset of \(V\) occurs in at most \(\lambda\) blocks. The packing problem is to determine the maximum number of blocks, \(\sigma(\nu,\kappa,\lambda)\), in a packing design. It is well known that \(\sigma(\nu,\kappa,\lambda) < \left[ \frac{\nu}{\kappa}[\frac{(\nu-1)}{\kappa(\kappa-1)}] \right] = \psi(\nu,\kappa,\lambda)\), where \([x]\) is the largest integer satisfying \(x \ge [x]\). It is shown here that if \(v \equiv 2 \pmod{4}\) and \(\nu \geq 6\) then \(\sigma(\nu,5,3) = \psi(\nu,5,3)\) with the possible exception of \(v = 38\).

In this paper we obtain some new relations on generalized exponents of primitive matrices. Hence the multiexponent of primitive tournament matrices are evaluated.

The ranking and unranking problem of a Gray code \(C(n,k)\) for compositions of \(n\) into \(k\) parts is solved. This means that rules have been derived by which one can calculate in a non-recursive way the index of a given codeword, and vice versa, determine the codeword with a given index. A number system in terms of binomial coefficients is presented to formulate these rules.

In the definition of local connectivity, the neighbourhood of a vertex consists of the induced subgraph of all vertices at distance one from the vertex. In {[2]}, we introduced the concept of distance-\(n\) connectivity in which the distance-\(n\) neighbourhood of a vertex consists of the induced subgraph of all vertices at distance less or equal to \(n\) from that vertex. In this paper we present Menger-type results for graphs whose distance-\(n\) neighbourhoods are all \(k\)-connected, \(n \geq 1\).

A partially ordered set \(P\) is called a circle order if one can assign to each element \(a \in P\) a circular disk in the plane \({C_a}\), so that \(a < b\) iff \(C_a \subset C_b\). It is known that the dual of every finite circle order is a circle order. We show that this is false for infinite circle orders.