Let \( G \) be a \((p,q)\)-graph where each edge of \( G \) is labeled by a number \( 1, 2, \ldots, q \) without repetition. The vertex sum for a vertex \( v \) is the sum of the labels of edges that are incident to \( v \). If the vertex sums are equal to a constant (mod \( k \)) where \( k \geq 2 \), then \( G \) is said to be Mod(\( k \))-edge-magic. In this paper, we investigate graphs which are Mod(\( k \))-edge-magic. When \( k = p \), the corresponding Mod(\( p \))-edge-magic graph is the edge-magic graph introduced by Lee (third author), Seah, and Tan in \([10]\). In this work, we investigate trees, unicyclic graphs, and \((p, p+1)\)-graphs which are Mod(2)-edge-magic.

First posed in 1942 by Kelly and Ulam, the Graph Reconstruction Conjecture is one of the major open problems in graph theory. While the Graph Reconstruction Conjecture remains open, it has spawned a number of related questions. In the classical vertex graph reconstruction number problem, a vertex is deleted in every possible way from a graph \( G \), and then it can be asked how many (both minimum and maximum) of these subgraphs are required to reconstruct \( G \) up to isomorphism. This can then be extended to deleting \( k \) vertices in every possible way.

Previous computer searches have found the 1-vertex-deletion reconstruction numbers of all graphs of up to 11 vertices. In this paper, computed values of \( k \)-vertex-deletion reconstruction numbers for all graphs on up to 8 vertices and \( k \leq |V(G)| – 2 \) are reported, as well as for some \( k \) for graphs on 9 vertices. Our data suggested a number of new theorems and conjectures. In particular, we pose, as a generalization of the Graph Reconstruction Conjecture, that any graph on \( 3k \) or more vertices is \( k \)-vertex-deletion reconstructible.

Let \( G \) be a simple graph with a vertex set \( V(G) \) and an edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces an edge partial labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = \lvert \{v \in V(G) : f(v) = i\} \rvert \) and \( e_f(i) = \lvert \{e \in E(G) : f^*(e) = i\} \rvert \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{\lvert e_f(0) – e_f(1) \rvert : \lvert v_f(0) – v_f(1) \rvert \leq 1\} \). In this paper, we investigate and present results concerning the balance index sets of trees of diameter four.

In this paper, we present new results about the coloring of graphs. We generalize the notion of proper vertex-coloring, introducing the concept of range-coloring of order \( k \). The relation between range-coloring of order \( k \) and total coloring is presented: we show that for any graph \( G \) that has a range-coloring of order \( \Delta(G) \) with \( t \) colors, there is a total coloring of \( G \) that uses \( (t+1) \) colors. This result provides a framework to prove that some families of graphs satisfy the total coloring conjecture. We exemplify with the family of block-cactus graphs.

We show, for \( k = 3, 4, 5 \), that the necessary conditions are sufficient for the existence of graph designs which decompose \( K_v(\lambda, j) \), the complete (multi)graph on \( v \) points with \( \lambda \) multiple edges for each pair of points and \( j \) loops at each vertex, into ordered blocks \( (a_1, a_2, \ldots, a_{k-1}, a_1) \). Each block is the subgraph which contains both the set of unordered edges \( \{a_i, a_j\} \), for each pair of consecutive edges in the ordered list, and also the loop at vertex \( a_1 \).

We investigate the existence of fixed point families for the eccentric digraph (\( \text{ED} \)) operator, which was introduced in \([1]\). In \([2]\), the notion of the period \( \rho(G) \) of a digraph \( G \) (under the \( \text{ED} \) operator) was defined, and it was observed, but not proved, that for any odd positive integer \( m \), \( C_m \times C_m \) is periodic, and that \( \rho(\text{ED}(C_m \times C_m)) = 2\rho(\text{ED}(C_m)) \). Also in \([2]\), the following question was posed: which digraphs are fixed points under the digraph operator? We provide a proof for the observations about \( C_m \times C_m \), and in the process show that these products comprise a family of fixed points under \( \text{ED} \). We then provide a number of other interesting examples of fixed point families.

In this paper, we derive some necessary existence conditions for balanced arrays (B-arrays) of strength eight and with two levels by making use of some classical inequalities such as Cauchy, Hölder, and Minkowski. We discuss the usefulness of these conditions in the study of the B-arrays, and also present some illustrative examples.

For a graph \( G \) having chromatic number \( k \), an equivalence relation is defined on the set \( X \) consisting of all proper vertex \( k \)-colorings of \( G \). This leads naturally to an equivalence relation on the set \( \mathcal{P} \) consisting of all partitions of \( V(G) \) into \( k \) independent subsets of color classes. The notion of a partition type arises and the algebra of types is investigated.

We derive a new upper bound of \( 26 \) for the Ramsey number \( R(K_5 – P_3, K_5) \), lowering the previous upper bound of \( 28 \). This leaves \( 25 \leq R(K_5 – P_5, K_5) \leq 26 \), improving on one of the three remaining open cases in Hendry’s table, which listed Ramsey numbers for pairs of graphs \( (G, H) \) with \( G \) and \( H \) having five vertices.

We also show, with the help of a computer, that \( R(B_2, B_6) = 17 \) and \( R(B_2, B_7) = 18 \) by full enumeration of \( (B_2, B_6) \)-\({good}\) graphs and \( (B_2, B_7) \)-\({good}\) graphs, where \( B_n \) is the book graph with \( n \) triangular pages.

Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). Any edge labeling \( f \) induces a partial vertex labeling \( f^+ : V(G) \to \mathbb{Z}_2 \) assigning \( 0 \) or \( 1 \) to \( f^+(v) \), \( v \) being an element of \( V(G) \), depending on whether there are more \( 0 \)-edges or \( 1 \)-edges incident with \( v \), and no label is given to \( f^+(v) \) otherwise. For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = \lvert \{v \in V(G) : f^+(v) = i\} \rvert \) and let \( e_f(i) = \lvert \{e \in E(G) : f(e) = i\} \rvert \). An edge-labeling \( f \) of \( G \) is said to be edge-friendly if \( \lvert e_f(0) – e_f(1) \rvert \leq 1 \). The edge-balance index set of the graph \( G \) is defined as \( \text{EBI}(G) = \{\lvert v_f(0) – v_f(1) \rvert : f \text{ is edge-friendly}\} \). In this paper, we investigate and present results concerning the edge-balance index sets of \( L \)-products of cycles with stars.