An elongated ply \( T(n; t^{(1)}, t^{(2)}, \ldots, t^{(n)}) \) is a snake of \( n \) number of plys \( P_{t(i)} (u_i, u_{i+1}) \) where any two adjacent plys \( P_{t(i)} \) and \( P_{t(i+1)} \) have only the vertex \( u_{i+1} \) in common. That means the block cut vertex graph of \( T_n \) is thus a path of length \( n – 1 \). In this paper, the cordiality of the Elongated Ply \( T_n \) is investigated.

Consider placing a guard on each vertex of a dominating set \( S_0 \) of a graph. If for every vertex \( v \notin S_0 \), there is a corresponding guard at an adjacent vertex \( u \) for which the resulting set \( S_1 = S_0 – \{u\} \cup \{v\} \) is dominating, then we say that \( S_0 \) is \( 1 \)-secure. It is eternally \( 1 \)-secure if for any sequence \( v_1, v_2, \ldots, v_k \) of vertices, there exists a sequence of guards \( u_1, u_2, \ldots, u_k \) with \( u_i \in S_{i-1} \) and \( u_i \) equal to or adjacent to \( v_i \), such that each set \( S_i = S_{i-1} – \{u_i\} \cup \{v_i\} \) is dominating. We investigate the minimum cardinality of an eternally secure set. In particular, we refute a conjecture of Burger et al. We also investigate eternal \( m \)-security, in which all guards can move simultaneously.

In 1990, Kolesova, Lam, and Thiel determined the 283,657 main classes of Latin squares of order 8. Using techniques to determine relevant Latin trades and integer programming, we examine representatives of each of these main classes and determine that none can contain a uniquely completable set of size less than 16. In three of these main classes, the use of trades which contain less than or equal to three rows, columns, or elements does not suffice to determine this fact. We closely examine properties of representatives of these three main classes. Writing the main result in Nelder’s notation for critical sets, we prove that \( \text{scs}(8) = 16 \).

An edge-ordering of a graph \( G = (V, E) \) is a one-to-one function \( f \) from \( E \) to the set of positive integers. A path of length \( k \) in \( G \) is called a \( (k, f) \)-ascent if \( f \) increases along the edge sequence of the path. The altitude \( \alpha(G) \) of \( G \) is the greatest integer \( k \) such that for all edge-orderings \( f \), \( G \) has a \( (k, f) \)-ascent.

We obtain a recursive lower bound for \( \alpha(K_{m,n}) \) and show that

\[\alpha(K_{3,n}) = \begin{cases}4 & \text{if } 5 \leq n \leq 9 \\5 & \text{if } 10 \leq n \leq 12 \\6 & \text{if } n \geq 13\end{cases}\]

A vertex set \( D \) of a graph \( G \) is a dominating set if every vertex not in \( D \) is adjacent to some vertex in \( D \). The domination number \( \gamma \) of a graph \( G \) is the minimum cardinality of a dominating set in \( G \). In 1989, Brigham and Dutton [1] proved

\[\gamma \leq \left\lceil\frac{3n-g}{6}\right\rceil\]

for each graph \( G \) of order \( n \), minimum degree \( \delta \geq 2 \), and girth \( g \geq 5 \). If \( G \) is a graph of order \( n \), minimum degree \( \delta \geq 2 \), girth \( g \geq 5 \) and neither a cycle nor one of two exceptional graphs, then we give in this paper the better bound

\[\gamma(G) \leq \left\lceil\frac{3n-g}{6}\right\rceil-1\]

For \( \delta \geq 3 \) and \( g \geq 5 \), we also prove \( \gamma \leq \left\lceil\frac{6n-g}{15}\right\rceil \), and this inequality is better than \( (*) \) when \( n > g + 10 \). In addition, if \( \delta \geq 3 \), then we show that

\[2\gamma \leq n – (\delta-2)(1 + \lfloor{d}/{3}\rfloor)\]

where \( d \) is the diameter of the graph. Some related bounds in terms of the diameter, girth, order, and minimum degree are also presented.

SOS-skeins correspond exactly to the Steiner quadruple systems [8,12]. Let \( P_1 \) be a finite simple SQS-skein of cardinality \( n > 4 \). In this article, we will present a construction for a non-simple subdirectly irreducible (monolithic) SOS-skein \( P = 2 \otimes_\alpha P_n \) of cardinality \( 2n \) in which each proper homomorphic image is Boolean for all \( n \equiv 2 \) or \( 4 \pmod{6} \). We can then show that if \( P_1 \) has a simple derived sloop, then the constructed SOS-skein \( 2 \otimes_\alpha P_1 \) contains a derived sloop which is subdirectly irreducible and has the same property as the SOS-skein \( 2 \otimes_\alpha P_1 \) that each of its proper homomorphic images is Boolean. Similar to the theory of Steiner loops and Steiner quasigroups [14], the author [1] has proven that the variety \( V(P_1) \) generated by a finite simple cubic SQS-skein \( P_1 \) covers the smallest non-trivial subvariety (the class of all Boolean SQS-skeins). Finally, we show that the variety \( V(2 \otimes_\alpha P_1) \) generated by the constructed SQS-skein \( 2 \otimes_\alpha P_1 \) covers the variety \( V(P_1) \) for each finite simple cubic SOS-skein \( P_1 \).

It is shown that for any rank \( r \) with \( n – \log(n+1) + 4 \leq r \leq n – 4 \) and any length \( n \), where \( n = 2^k – 1 \) and \( k \geq 8 \), there is a perfect code with these parameters and with a trivial group of symmetries.

In this paper, we introduce, for the first time, the notion of self-dual modular-graceful labeling of a cyclic digraph. A cyclic digraph \( G(V, E) \) is a digraph whose connected components are directed cycles. The line digraph \( G^\wedge(V^\wedge, E^\wedge) \) of the cyclic digraph \( G \) is the digraph where \( V^\wedge = E \), \( E^\wedge = V \), and if \( \alpha, \beta \) are two edges of \( G \) which join vertex \( x \) to vertex \( y \) and vertex \( y \) to vertex \( z \) respectively, then in the digraph \( G^\wedge \), \( y \) is the edge joining vertex \( \alpha \) to vertex \( \beta \). A labeling \( f \) for a cyclic digraph of order \( n \) is a map from \( V \) to \( \mathbb{Z}_{n+1} \). The labeling \( f \) induces a dual labeling \( f^\wedge \) for \( G^\wedge \) by \( f^\wedge(\alpha) = f(x) – f(y) \), where \( \alpha \) is an edge of \( G \) which joins vertex \( x \) to vertex \( y \). A self-dual modular-graceful cyclic digraph \( G \) is a cyclic digraph together with a labeling \( f \) where the image \( f(V) = \mathbb{Z}_{n+1}^* \), and \( \langle G^\wedge, f^\wedge \rangle \) is an isomorphic digraph of \( \langle G, f \rangle \). We prove the necessary and sufficient conditions for the existence of self-dual modular-graceful cyclic digraphs and connected self-dual modular-graceful cyclic digraphs. We also give some explicit constructions of these digraphs in the case \( n+1 \) is prime and in the general case where \( n+1 \) is not prime.

We refer to a labeling of a plane graph as a \( d \)-antimagic labeling if the vertices, edges, and faces of the graph are labeled 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 form an arithmetic progression of difference \( d \). This paper describes \( d \)-antimagic labelings for a special class of plane graphs.

2-trees are defined recursively, starting from a single edge, by repeatedly erecting new triangles onto existing edges. These have been widely studied in connection with chordal graphs, series-parallel graphs, and isolated failure immune (IFI) networks.

A similar family, based on recursively erecting new \( K_{2,h} \) subgraphs onto existing edges, is shown to have analogous connections to chordal bipartite graphs, series-parallel graphs, and a notion motivated by IFI networks.