In this paper, we show that for every sufficiently large integer \(n\) and every positive integer \(c \leq \left\lfloor \frac{1}{6}({\log \log n})^\frac{1}{2} \right \rfloor\), a Boolean lattice with \(n\) atoms can be partitioned into chains of cardinality \(c\), except for at most \(c-1\) elements which also form a chain.

We construct all self-dual \([24, 12, 8]\) quaternary codes with a monomial automorphism of prime order \(r > 3\) and obtain a unique code for \(r = 23\) (which has automorphisms of orders \(5\), \(7\), and \(11\) too), two inequivalent codes for \(r = 11\), \(6\) inequivalent codes for \(r = 7\), and \(12\) inequivalent codes for \(r = 5\). The obtained codes have \(12\) different weight spectra.

Metamorphoses of small \(k\)-wheel systems for \(k = 3, 4,\) and \(6\) are obtained. In particular, we obtain simultaneous metamorphoses of: \(3\)-wheel systems into Steiner triple systems and into \(K_{1,3}\)-designs; \(4\)-wheel systems into \(4\)-cycle systems, \(K_{1,4}\)-designs, and bowtie systems; \(6\)-wheel systems into \(6\)-cycle systems, \(K_{1,6}\)-designs, and \(3\)-windmill designs or near-\(3\)-windmill designs.

We deal with the problem of labeling the vertices, edges, and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all the faces constitute an arithmetical progression of difference \(d\).

If \(L\) is a list assignment function and \(\kappa\) is a multiplicity function on the vertices of a graph \(G\), a certain condition on \((G, L, \kappa)\), known as Hall’s multicoloring condition, is obviously necessary for the existence of a multicoloring of the vertices of \(G\). A graph \(G\) is said to be in the class \(MHC\) if it has a multicoloring for any functions \(L\) and \(\kappa\) such that \((G, L, \kappa)\) satisfies Hall’s multicoloring condition. It is known that if \(G\) is in \(MHC\) then each block of \(G\) is a clique and each cutpoint lies in precisely two blocks. We conjecture that the converse is true as well. It is also known that if \(G\) is a graph consisting of two cliques joined at a point then \(G\) is in \(MHC\). We present a new proof of this result which uses common partial systems of distinct representatives, the relationship between matching number and vertex covering number for 3-partite hypergraphs, and Menger’s Theorem.

This paper presents a new approach in the quest for a solution to the \(3x+1\) problem. The method relies on the convergence of the trajectories of the odd positive integers by exploiting the role of the positive integers of the form \(1+4n\), where \(n\) is a non-negative integer.