Let \(G\) be a group acting on a set \(\Omega\). A subset (finite or infinite) \(A \subseteq \Omega\) is called \(k\)-quasi-invariant, where \(k\) is a non-negative integer, if \(|A^g \backslash A| \leq k\) for every \(g \in G\). In previous work of the authors a bound was obtained, in terms of \(k\), on the size of the symmetric difference between a \(k\)-quasi-invariant subset and the \(G\)-invariant subset of \(\Omega\) closest to it. However, apart from the cases \(k = 0, 1\), this bound gave little information about the structure of a \(k\)-quasi-invariant subset. In this paper a classification of \(2\)-quasi-invariant subsets is given. Besides the generic examples (subsets of \(\Omega\) which have a symmetric difference of size at most \(2\) with some \(G\)-invariant subset) there are basically five explicitly determined possibilities.

This note is an extension of \([4]\) , wherein is shown a relation between the dual notions of graceful and edge-graceful graphs. In particular, this note proves two graceful conjectures raised in \([4]\) , and then utilizes the result to edge-gracefully label certain trees not previously known to be edge-graceful.

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.