We prove that the smallest covering code of length \(8\) and covering radius \(2\) has exactly \(12\) words. The proof is based on partial classification of even weight codewords, followed by a search for small sets of odd codewords covering the part of the space that has not been covered by the even subcode.

Alon and Yuster {[4]} have proven that if a fixed graph \(K\) on \(g\) vertices is \((h+1)\)-colorable, then any graph \(G\) with \(n\) vertices and minimum degree at least \(\frac{h}{h+1}n\) contains at least \((1-\epsilon)\frac{n}{g})\) vertex disjoint copies of \(K\), provided \(n>N(\epsilon)\). It is shown here that the required minimum degree of \(G\) for this result to follow is closer to \(\frac{h-1}{h }n\), provided \(K\) has a proper \((h+1)\)-coloring in which some of the colors occur rarely. A conjecture regarding the best possible result of this type is suggested.

Let \(G\) be a finite group with a normal subgroup \(H\). We prove that if there exist a \((h, r;\lambda, H)\) difference matrix and a \((g/h, r;1, G/H)\) difference matrix, then there exists a \((g, r;\lambda, G)\) difference matrix. This shows in particular that if there exist \(r\) mutually orthogonal orthomorphisms of \(H\) and \(r\) mutually orthogonal orthomorphisms of \(G/H\), then there exist \(r\) mutually orthogonal orthomorphisms of \(G\). We also show that a dihedral group of order \(16\) admits at least \(3\) mutually orthogonal orthomorphisms.

Let \(k\) and \(b\) be integers and \(k > 1\). A set \(S\) of integers is called \((k, b)\) linear-free (or \((k, b)\)-LF for short) if \(2 \in S\) implies \(kx + b \notin S\). Let \(F(n, k, b) = \max\{|A|: A \text{ is } (k, 0)\text{-LF and } A \subseteq [1, n]\}\), where \([1, n]\) denotes all integers between \(1\) and \(n\). A subset \(A\) of \([1, n]\) with \(|A| = F(n, k, b)\) is called a maximal \((k, b)\)-LF subset of \([1, n]\). In this paper, a recurrence relation for \(F(n, k, b)\) is obtained and a method to construct a maximal \((k, b)\)-LF subset of \([1, n]\) is given.

This paper deals with a new kind of graph labeling similar to the well known harmonious, graceful, and elegant labelings. A polychrome labeling of a simple and connected graph \(G = (V, E)\) by an abelian group \(A\) is a bijective map from \(V\) onto \(A\) such that the induced edge labeling \(f^*(uv) = f(v) + f(w), uv \in E\), is injective. Polychrome labelings of a path and a cycle by a large class of abelian groups are designed, and the connection to the above mentioned labelings is shown. In addition, the author presents a conjecture which is similar to a famous conjecture of G. Ringel about graceful trees (see {[9]}).

A graph is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore this cardinality is exactly half the number of vertices, the graph is called very well covered. Sankaranarayana in \({[5]}\) presented a certain subclass of well covered graphs (called Wan) and gave a characterization of this class which generalized the characterization of very well covered graphs given by Favaron \([2]\) . The purpose of this article is to generalize to this new subclass some results concerning the stability, domination, and irredundance parameters proved for very well covered graphs in \([2]\) .

Three new characterizations of matroids are presented.

A decomposition of a graph \(H\) is a family of subgraphs of \(H\) such that each edge of \(H\) is contained in exactly one member of the family. For a graph \(G\), a \(G\)-decomposition of the graph \(H\) is a decomposition of \(H\) into subgraphs isomorphic to \(G\). If \(H\) has a \(G\)-decomposition, \(H\) is said to be \(G\)-decomposable; this is denoted by \(H \rightarrow G\). In this paper, we prove by construction that the complete graph \(K_{24}\) is \(G\)-decomposable, where \(G\) is the complementary graph of the path \(P_5\).

A unified approach to prove former connectivity results of Tutte, Cunningham, Inukai, and Weinberg, Oxley, and Wagner.