In the \(n\)-dimensional hypercube, an \(n\)-snake is a simple path with no chords, while an \(n\)-coil is a simple cycle without chords. There has been much interest in determining the length of a maximum \(n\)-snake and a maximum \(n\)-coil. Only upper and lower bounds for these maximum lengths are known for arbitrary \(n\). Computationally, the problem of finding maximum \(n\)-snakes and \(n\)-coils suffers from combinatorial explosion, in that the size of the solution space which must be searched grows very rapidly as \(n\) increases. Previously, the maximum lengths of \(n\)-snakes and \(n\)-coils have been established only for \(n \leq 7\)and \(n \leq 6\), respectively. In this paper, we report on a coil searching computer program which established that \(48\) is the maximum length of a coil in the hypercube of dimension \(7\).

The complements of the perfect dominating sets of the \(n\)-cube, for \(n \leq 8\), are characterized as well as some outstanding vertex-spanning edge-partitions of them involving the Fano plane, as a contribution to the study of distance-preserving regular subgraphs of hypercubes.

A graph is said to be in \({L}_1\) if \(\deg(u) + \deg(v) \geq |N(u) \cup N(w) \cup N(v)| – 1\) for each induced path \(uwv\) of order three. We prove that a \(2\)-connected graph \(G\) in \({L}_1\) of diameter two is hamiltonian, or \(K_{d,d+1} \subset G \subset K_{d} + (d + 1)K_1\) for some \(d \geq 2\). This theorem generalizes a couple of known sufficient conditions for a graph to be hamiltonian. We also discuss the relation between this theorem and several other degree conditions for hamiltonicity.

On the basis of circuit uniqueness, the concept of strong circuit uniqueness is introduced, and some graphs with the property of strong circuit uniqueness are identified. The results are then used to prove successfully the circuit uniqueness of the graphs \(K_m \cup K_n\) and \(K_{m,n}\). This represents an improvement on the previous papers on the same subject.

Several criteria have been proposed as desirable for binary cryptographic functions. Three important ones are balance, correlation-immunity, and higher order strict avalanche criterion. Lloyd [7] has shown that there are no balanced, uncorrelated functions which satisfy the strict avalanche criterion of order \(n-2\). In this note, we give a short proof of this result using elementary combinatorial arguments. The proof relies on the solution of a recurrence relation that seems to be of interest in its own right.

In this paper, we introduce some concepts relating to idempotent ordered orthogonal quasigroups (IOOQ), ordered orthogonal Steiner triple systems (ordered OSTS), and ordered orthogonal group divisible designs (ordered OGDD), and use them to obtain some construction methods for OGDD.

It is known that triangle-free graphs of diameter \(2\) are just maximal triangle-free graphs. Kantor ([5]) showed that if \(G\) is a triangle-free and \(4\)-cycle free graph of diameter \(2\), then \(G\) is either a star or a Moore graph of diameter \(2\); if \(G\) is a \(4\)-cycle free graph of diameter \(2\) with at least one triangle, then \(G\) is either a star-like graph or a polarity graph (defined from a finite projective plane with polarities) of order \(r^2 + r + 1\) for some positive integer \(r\) (or \(P_r\)-\({graph}\) for short). We study, by purely graph theoretical means, the structure of \(P_r\)-graphs and construct \(P_r\)-graphs for small values of \(r\). Further, we characterize graphs of diameter \(2\) without \(5\)-cycles and \(6\)-cycles, respectively. In general, one can characterize \(C_k\)-free graphs of diameter \(2\) with \(k > 6\) with a similar approach.

Dey’s formula can be used to count the subgroups of finitely generated groups and to establish congruence properties of subgroup counting functions. We develop an algebraic technique based on this formula for counting the subgroups of given index in Hecke groups, and show how to streamline it for efficient computation modulo \(2\).

A simple graph \(G\) with a perfect matching is said to be \({k-extendable}\) if for every set \(M\) of \(k\) independent edges, there exists a perfect matching in \(G\) containing all the edges of \(M\). In an earlier paper, we characterized \((n-2)\)-extendable graphs on \(2n \geq 10\) vertices. In this paper, we complete the characterization by resolving the remaining small cases of \(2n = 6\) and \(8\). In addition, the subclass of \(k\)-extendable graphs that are “critical” and “minimal” are determined.