Let \( G \) be the one-point union of two cycles and suppose \( G \) has \( n \) edges. We show via various graph labelings that there exists a cyclic \( G \)-decomposition of \( K_{2nt+1} \) for every positive integer \( t \).

Recently Ozbal and Firat [22] introduced the notion of symmetric \( f \) bi-derivation of a lattice. They give illustrative examples and they also characterized the distributive lattice by symmetric \( f \) bi-derivation. In this paper, we define the isotone symmetric \( f \) bi-derivation and obtain some interesting results about isotoneness. We also provide the relations between distributive, modular, and isotone lattices through symmetric \( f \) bi-derivation.

In 2003, Lee, Wang and Wen found a non-edge-magic simple connected cubic graph which satisfying the necessary condition of edge-magicness by using computer search. They asked for a mathematical proof. In this paper, we will provide such a proof.

Let \( G \) be a graph and let \( f \) be a positive integer-valued function defined on \( V(G) \) such that \( 1 \leq a \leq f(x) \leq b \leq 2a \) for every \( x \in V(G) \). If \( t(G) \geq \frac{b^2}{a} \), \( |V(G)| \geq \frac{b^2}{a} + 1 \), and \( f(V(G)) \) is even, then \( G \) has an \( f \)-factor.

A general construction for \( t \)-SB(\(2t-1\), \(2t-2\)) designs is given. In addition, large sets of \( t \)-SB(\(v\), \(k\)) are discussed and some examples are provided.

For a poset \( P = (X, \leq_P) \), the strict-double-bound graph (\(sDB\)-graph) of \( P = (X, \leq_P) \) is the graph \( sDB(P) \) on \( X \) for which vertices \( u \) and \( v \) of \( sDB(P) \) are adjacent if and only if \( u \neq v \) and there exist \( x \) and \( y \) in \( X \) distinct from \( u \) and \( v \) such that \( x \leq u \leq y \) and \( x \leq v \leq y \). The strict-double-bound number \( \zeta(G) \) is defined as

\[
\zeta(G) = \min \{ n \mid G \cup N_n \text{ is a strict-double-bound graph} \},
\]

where \( N_n \) is the graph with \( n \) vertices and no edges.

In this paper we deal with strict-double-bound numbers of some graphs. For example, we obtain that

\[
\zeta(P_n) = \lceil 2\sqrt{n-1} \rceil \text{ (} n \geq 2 \text{)},
\]

\[
\zeta(C_n) = \lceil 2\sqrt{n} \rceil \text{ (} n \geq 4 \text{)},
\]

\[
\zeta(W_n) = \lceil 2\sqrt{n-1} \rceil \text{ (} n \geq 5 \text{)},
\]

and

\[
\zeta(G + K_n) = \zeta(G)
\]

for a graph \( G \) with no isolated vertices.

The metric dimension of a graph \(G\), denoted by \(\text{dim}(G)\), is the minimum number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. For a graph \(G\) and its complement \(\overline{G}\), each of order \(n \geq 4\) and connected, we show that

\[
2 \leq \text{dim}(G) + \text{dim}(\overline{G}) \leq 2(n-3).
\]

It is readily seen that \(\text{dim}(G) + \text{dim}(\overline{G}) = 2\) if and only if \(n = 4\). We characterize graphs satisfying

\[
\text{dim}(G) + \text{dim}(\overline{G}) = 2(n-3)
\]

when \(G\) is a tree or a unicyclic graph.

Whist tournament designs are known to exist for all \( v \equiv 0,1 \pmod{4} \). Much less is known about the existence of \(\mathbb{Z}\)-cyclic whist designs. Previous studies \([5, 6]\) have reported on all \(\mathbb{Z}\)-cyclic whist designs for \( v \in \{4,5,8,9,12,13,16,17,20,21,24,25\} \). This paper is a report on all \(\mathbb{Z}\)-cyclic whist tournament designs on 28 players, including a detailed summary of all known whist specializations related to a 28 player \(\mathbb{Z}\)-cyclic whist design. Our study shows that there are \( 7,910,127 \) \(\mathbb{Z}\)-cyclic whist designs on 28 players. Of these designs, \( 2,568,510 \) possess the Three Person Property, \( 240,948 \) possess the Triplewhist Property and none possess the Balancedwhist Property. Introduced here is the concept of the mirror image of a \(\mathbb{Z}\)-cyclic whist design. In general, utilization of this concept reduces the computer search for \(\mathbb{Z}\)-cyclic whist designs by nearly fifty percent.

Let \( S \) be an orthogonal polygon in the plane bounded by a simple closed curve. Assume that every two boundary points of \( S \) have a common staircase illuminator whose edges are north and east. Then \( S \) contains a staircase path \( \mu_0 \) whose edges are north and east such that \( \mu_0 \) illumines every point of \( S \). Without the requirement that the illuminators share a common direction, the result fails.