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.

The upper domination Ramsey number \(u(3, 3, 3)\) is the smallest integer \(n\) such that every \(3\)-coloring of the edges of complete graph \(K_n\) contains a monochromatic graph \(G\) with \(\Gamma(\overline{G}) \geq 3\), where \(\Gamma(\overline{G})\) is the maximum order over all the minimal dominating sets of the complement of \(G\). In this note, with the help of computers, we determine that \(U(3, 3, 3) = 13\), which improves the results that \(13 \leq U(3, 3, 3) \leq 14\) provided by Michael A. Henning and Ortrud R. Oellermann.

Let \(G\) be a graph of order \(n \geq 4k+8\), where \(k\) is a positive integer with \(kn\) even and \(\delta(G) > k+1\). We show that if \(max\{d_G(u),d_G(v)\} > {n}/{2}\) for each pair of nonadjacent vertices \(u,v\), then \(G\) has a connected \([k, k+1]\)-factor excluding any given edge \(e\).

A \( k \)-edge labeling of a graph \( G \) is a function \( f \) from the edge set \( E(G) \) to the set of integers \( \{0, \ldots, k-1\} \). Such a labeling induces a labeling \( f \) on the vertex set \( V(G) \) by defining \( f(v) = \sum f(e) \), where the summation is taken over all the edges incident on the vertex \( v \) and the value is reduced modulo \( k \). Cahit calls this labeling edge-\( k \)-equitable if \( f \) assigns the labels \( \{0, \ldots, k-1\} \) equitably to the vertices as well as edges.

If \( G_1, \ldots, G_T \) is a family of graphs having a graph \( H \) as an induced subgraph, then by \( H \)-union \( G \) of this family we mean the graph obtained by identifying all the corresponding vertices as well as edges of the copies of \( H \) in \( G_1, \ldots, G_T \).

In this paper we prove that the \( \overline{K}_n \)-union of gears is edge-\( 3 \)-equitable.

Let \( k \) be a positive integer and let \( G \) be a simple graph with vertex set \( V(G) \). If \( v \) is a vertex of \( G \), then the open \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}(v) \), is the set \( N_{k,G}(v) = \{u \mid u \neq v \text{ and } d(u, v) \leq k\} \). The closed \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}[v] \), is \( N_{k,G}[v] = N_{k,G}(v) \cup \{v\} \). A function \( f: V(G) \to \{-1,1\} \) is called a \({signed\; distance \; k -dominating\; function}\) if \( \sum_{u \in N_{k,G}(v)} f(u) \geq 1 \) for each vertex \( v \in V(G) \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed distance \( k \)-dominating functions on \( G \) with the property that \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V(G) \) is called a \({signed\; distance \; k -dominating \;family}\) (of functions) on \( G \). The maximum number of functions in a signed distance \( k \)-dominating family on \( G \) is the \({signed\; distance \; k -domatic\; number}\) of \( G \), denoted by \( d_{k,s}(G) \). Note that \( d_{1,s}(G) \) is the classical signed domatic number \( d_s(D) \). In this paper, we initiate the study of signed distance \( k \)-domatic numbers in graphs and we present some sharp upper bounds for \( d_{k,s}(G) \).

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacency matrix and automorphism group.

In this paper, we consider a variation of toughness, and prove stronger results for the existence of \([a, b]\)-factors. Furthermore, we show that the results are sharp in some sense.

A decomposition \( \mathcal{D} \) of a graph \( H \) by a graph \( G \) is a partition of the edge set of \( H \) such that the subgraph induced by the edges in each part of the partition is isomorphic to \( G \). The intersection graph \( I(\mathcal{D}) \) of the decomposition \( \mathcal{D} \) has a vertex for each part of the partition and two parts \( A \) and \( B \) are adjacent if and only if they share a common node in \( H \). If \( I(\mathcal{D}) \cong H \), then \( \mathcal{D} \) is an automorphic decomposition of \( H \). If \( n(G) = \chi(H) \) as well, then we say that \( \mathcal{D} \) is a fully automorphic decomposition. In this paper, we examine the question of whether a fully automorphic host will have an even degree of regularity. We also give several examples of fully automorphic decompositions as well as necessary conditions for their existence.