Let $o\left(n\right)$ be the greatest odd integer less than or equal to $n$. In this paper we provide explicit formulae to compute $\mathbb{N}$-graded Betti numbers of the circulant graphs $C_{2n}(1,2,3,5,...,o\left(n\right))$. We do this by showing that this graph is the product (or join) of the cycle $C_n$ by itself, and computing Betti numbers of $C_n*C_n$. We also discuss whether such a graph (more generally, $G*H$) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or $S_2$.

Using results of Altshuler and Negami, we present a classification of biembeddings of symmetric configurations of triples in the torus or Klein bottle. We also give an alternative proof of the structure of 3-homogeneous Latin trades.

The following result is proved: if a bipartite graph is not a circle graph, then its complement is not a circle graph. The proof uses Naji’s characterization of circle graphs by means of a linear system of equations with unknowns in $\mathrm{GF}\left(2\right)$.At the end of this short note I briefly recall the work of François Jaeger on circle graphs.

Let $G$ be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that $G$ admits a bipartition such that each vertex class meets edges of total weight at least $(w_1-{\Delta }_1)/2+2w_2/3$, where $w_i$ is the total weight of edges of size $i$ and ${\Delta }_1$ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph $G$ (i.e., multi-hypergraph), we show that there exists a bipartition of $G$ such that each vertex class meets edges of total weight at least $(w_0-1)/6+(w_1-{\Delta }_1)/3+2w_2/3$, where $w_0$ is the number...