In this paper, necessary and sufficient conditions are given for the existence of extended \(5\)-cycle systems of order \(n\) which have \(r\) idempotent elements.

An \((f,2)\)-graph is a multigraph \(G\) such that each vertex of \(G\) has degree either \(f\) or \(2\). Let \(S(n, f)\) denote the simple graph whose vertex set is the set of unlabeled \((f,2)\)-graphs of order no greater than \(n\) and such that \(\{G, H\}\) is an edge in \(S(n, f)\) if and only if \(H\) can be obtained from \(G\) by either an insertion or a suppression of a vertex of degree \(2\). We also consider digraphs whose nodes are labeled or unlabeled \((f, 2)\)-multigraphs and with arcs \((G, H)\) defined as for \(\{G, H\}\).

We study the structure of these graphs and digraphs. In particular, the diameter of a given component is determined. We conclude by defining a random process on these digraphs and derive some properties. Chemistry applications are suggested.

Given a coloring \(f\) of Euclidean space \(\mathbb{R}^n\) and some group \(G\) of its transformations, its subsets \(A\) and \(B\) are said to be colored similarly, if there exists \(g \in G\), such that \(B = g(A)\) and \(f(a) = f(g(a))\), for all \(a \in A\). From our earlier result [12] it follows that there are \(2\)-colorings of \(\mathbb{R}^n\), in which no two different line segments are colored similarly with respect to isometries. The main purpose of this paper is to investigate other types of such pattern avoiding colorings. In particular, we consider topological as well as measure theoretic aspects of the above scene. Our motivation for studying this topic is twofold. One is that it extends square-free colorings of \(\mathbb{R}\), introduced in [2] as a continuous version of the famous non-repetitive sequences of Thue. The other is its relationship to some exciting problems and results of Euclidean Ramsey Theory, especially those concerning avoiding distances.

In this paper, a definition of perfect binary matroids is considered and it is shown that, analogous to the Perfect Graph Theorem of Lovász and Fulkerson, the complement of a perfect matroid is also a perfect matroid. In addition, the classes of critically imperfect graphic matroids and critically imperfect graphs are compared.

A \((p,q)\) graph \(G\) is edge-magic if there exists a bijective function \(f : V(G) \cup E(G) \to \{1,2,\ldots,p+q\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant, called the valence of \(f\), for any edge \(uv\) of \(G\). Moreover, \(G\) is said to be super edge-magic if \(f(V(G)) = \{1,2,\ldots,p\}\). Every super edge-magic \((p,q)\) graph is cordial, and it is harmonious and sequential whenever it is a tree or \(q \geq p\). In this paper, it is shown to be edge-antimagic as well. The super edge-magic properties of several classes of connected and disconnected graphs are studied. Furthermore, we prove that there can be arbitrarily large gaps among the possible valences for certain super edge-magic graphs. We also establish that the disjoint union of multiple copies of a super edge-magic linear forest is super edge-magic if the number of copies is odd.

In this paper, necessary and sufficient conditions are given for the metamorphosis of a \(\lambda\)-fold \(K_{3,3}\)-design of order \(n\) into a \(\lambda\)-fold \(6\)-cycle system of order \(n\), by retaining one \(6\)-cycle subgraph from each copy of \(K_{3,3}\), and then rearranging the set of all the remaining edges, three from each \(K_{3,3}\), into further \(6\)-cycles so that the result is a \(\lambda\)-fold \(6\)-cycle system.

Partially balanced diallel cross block designs with \(m\) associate classes are defined and two general methods of construction are presented. Two-associate class designs based upon group divisible, triangular, and extended group divisible association schemes obtained using the general methods are also given. Tables of designs for no more than \(24\) parental lines are provided.

Given a non-planar graph \(G\) with a subdivision of \(K_5\) as a subgraph, we can either transform the \(K_5\)-subdivision into a \(K_{3,3}\)-subdivision if it is possible, or else we obtain a partition of the vertices of \(G\backslash K_5\) into equivalence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a \(K_{3,3}\)-subdivision in the graph \(G\). It significantly simplifies algorithms presented in [7], [10], and [12]. We then need to consider only the embeddings on the given surface of a \(K_{3,3}\)-subdivision, which are much less numerous than those of \(K_5\).

Let \(M(d,n)\) denote the minimax number of group tests required for the identification of the \(d\) defectives in a set of \(n\) items. It was conjectured by Hu, Hwang, and Wang that \(M(d,n) = n-1\) for \(n \leq 3d\), a surprisingly difficult combinatorial problem with very little known. The best known result is \(M(d,n) = n-1\) for \(n \leq \frac{42}{16}d\) by Du and Hwang. In this note, we improve their result by proving \(M(d,n) = n – 1\) for \(d \geq 193\) and \(n \leq \frac{42}{16}d\).

In this paper, we investigate the divisibility of \(mn\) by \(am+bn+c\) for given \(a\), \(b\), and \(c\). We give the necessary and sufficient condition for the divisibility, that is, \(am + bn + c\) divides \(mn\). We then present the structure of the set of pairs \([m,n]\) that satisfies the divisibility. This structure is represented by a directed graph and we prove the necessary and sufficient condition for the graph to have a binary tree structure. In particular, for \(c = -1\), we show double binary tree structures on the set.