A \(4\)-regular graph \(G\) is called a \(4\)-circulant if its adjacency matrix \(A(G)\) is a circulant matrix. Because of the special structure of the eigenvalues of \(A(G)\), the rank of such graphs is completely determined. We show how all disconnected \(4\)-circulants are made up of connected \(4\)-circulants and classify all connected \(4\)-circulants as isomorphic to one of two basic types.

Let \([n, k, d; g]\)-codes be linear codes of length \(n\), dimension \(k\) and minimum Hamming distance \(d\) over \(\mathrm{GF}(g)\). Let \(d_8(n, k)\) be the maximum possible minimum Hamming distance of a linear \([n, k, d; 8]\)-code for given values of \(n\) and \(k\). In this paper, twenty-two new linear codes over \(\mathrm{GF}(8)\) are constructed which improve the bounds on \(d_8(n, k)\).

We find new full orthogonal designs in order \(56\) and show that of
\(1285\) possible \(OD(56; s_1, s_2, s_3,56 – s_1 – s_2 – s_3)\) \(163\) are known, of
\(261\) possible \(OD(56; s_1, s_2, 56 – s_1 – s_2)\) \(179\) are known. All possible
\(OD(56; s_1,56 – s_1)\) are known.

Sattolo has presented an algorithm to generate cyclic permutations at random. In this note, the two parameters “number of moves” and “distance” are analyzed.

In this paper, we shall classify the self-complementary graphs with minimum degree exactly \(2\).

A graphical partition of the even integer \(n\) is a partition of \(n\) where each part of the partition is the degree of a vertex in a simple graph and the degree sum of the graph is \(n\). In this note, we consider the problem of enumerating a subset of these partitions, known as graphical forest partitions, graphical partitions whose parts are the degrees of the vertices of forests (disjoint unions of trees). We shall prove that

\[gf(2k) = p(0) + p(1) + p(2) + \cdots + p(k-1)\]

where \(g_f(2k)\) is the number of graphical forest partitions of \(2k\) and \(p(j)\) is the ordinary partition function which counts the number of integer partitions of \(j\).

We make further progress towards the forbidden-induced-subgraph characterization of the graphs with Hall number \(\leq 2\). We solve several problems posed in [4] and, in the process, describe all “partial wheel” graphs with Hall number \(\geq 2\) with every proper induced subgraph having Hall number \(\leq 2\).

A radio labeling of a connected graph $G$ is an assignment of distinct, positive integers to the vertices of \(G\), with \(x \in V(G)\) labeled \(c(x)\), such that

\[d(u, v) + |c(u) – c(v)| \geq 1 + diam(G)\]

for every two distinct vertices \(u,v\) of \(G\), where \(diam(G)\) is the diameter of \(G\). The radio number \(rn(c)\) of a radio labeling \(c\) of \(G\) is the maximum label assigned to a vertex of \(G\). The radio number \(rn(G)\) of \(G\) is \(\min\{rn(c)\}\) over all radio labelings \(c\) of \(G\). Radio numbers of cycles are discussed and upper and lower bounds are presented.

Dudeney’s round table problem was proposed about one hundred years ago. It is already solved when the number of people is even, but it is still unsettled except for only a few cases when the number of people is odd.

In this paper, a solution of Dudeney’s round table problem is given when \(n = p+2\), where \(p\) is an odd prime number such that \(2\) is the square of a primitive root of \(\mathrm{GF}(p)\), \(p \equiv 1 \pmod{4}\), and \(3\) is not a quadratic residue modulo \(p\).