A diagonalised lattice is a two dimensional grid, where we add exactly one arbitrary diagonal in each square, and color each vertex black or white.We show that for every diagonalised lattice there is a walk from the left to the right, using only black vertices, if and only if there is no walk from the top to the bottom, using only white vertices.

A terrace for \(\mathbb{Z}_m\) is an arrangement \((a_1, a_2, \ldots, a_m)\) of the \(m\) elements of \(\mathbb{Z}_m\) such that the sets of differences \(a_{i+1} – a_i\) and \(a_i – a_{i+1}\) (\(i = 1, 2, \ldots, m-1\)) between them contain each element of \(\mathbb{Z}_m \setminus \{0\}\) exactly twice. For \(m\) odd, many procedures are available for constructing power-sequence terraces for \(\mathbb{Z}_m\); each such terrace may be partitioned into segments one of which contains merely the zero element of \(\mathbb{Z}_m\) whereas each other segment is either (a) a sequence of successive powers of a non-zero element of \(\mathbb{Z}_m\) or (b) such a sequence multiplied throughout by a constant. For \(n\) an odd prime power satisfying \(n \equiv 1\) or \(3 \pmod{8}\), this idea has previously been extended by using power-sequences in \(\mathbb{Z}_n\) to produce some \(\mathbb{Z}_m\) terraces \((a_1, a_2, \ldots, a_m)\) where \(m = n+1 = 2^\mu\), with \(a_{i+1} – a_i = -(a_{i+1+\mu} – a_{i+\mu})\) for all \(i \in [1, \mu-1]\). Each of these “da capo directed terraces” consists of a sequence of segments, one containing just the element \(0\) and another just containing the element \(n\), the remaining segments each being of type (a) or (b) above with each of its distinct entries \(z\) from \(\mathbb{Z}_n \setminus \{0\}\) evaluated so that \(1 \leq x \leq n-1\). Now, for many odd prime powers \(n\) satisfying \(n \equiv 1 \pmod{4}\), we similarly produce narcissistic terraces for \(\mathbb{Z}_{n+1}\); these have \(a_{i+1} – a_i = a_{m-i+1} – a_{m-i}\) for all \(i \in [1, \mu-1]\).

We determine the full Sylow \(p\)-subgroup of the automorphism group of transitive \(k\)-ary relational structures of order \(p^2\), \(p\) a prime. We then find the full automorphism group of transitive ternary relational structures of order \(p^2\), for those values of \(p\) for which \({A_p}\) is the only doubly-transitive nonabelian simple group of degree \(p\). Finally, we determine optimal necessary and sufficient conditions for two Cayley \(k\)-ary relational structures of order \(p^2\), \(k < p\), to be isomorphic.

Let \(G\) be a finite group, \(S\) (possibly, contains the identity element) be a subset of \(G\). The Bi-Cayley graph \(\text{BC}(G, S)\) is a bipartite graph with vertex set \(G \times \{0,1\}\) and edge set \(\{\{(g,0), (gs,1)\}, g \in G, s \in S\}\). A graph \(X\) is said to be super-edge-connected if every minimum edge cut of \(X\) is a set of edges incident with some vertex. The restricted edge connectivity \(\lambda'(X)\) of \(X\) is the minimum number of edges whose removal disconnects \(X\) into nontrivial components. A \(k\)-regular graph \(X\) is said to be optimally super-edge-connected if \(X\) is super-edge-connected and its restricted edge connectivity attains the maximum \(2k-2\). In this paper, we show that all connected Bi-Cayley graphs, except even cycles, are optimally super-edge-connected.