We give an upper bound on the number of edges of a graph with \( n \) vertices to be a prime cordial graph, and we improve this upper bound to fit bipartite graphs. Also, we determine all prime cordial graphs of order \( \leq 6 \).

We consider the one-color graph avoidance game. Using a high-performance computing network, we showed that the first player can win the game on \( 13 \), \( 14 \), and \( 15 \) vertices. Other related games are also discussed.

Let \( G \, \Box \, H \) denote the Cartesian product of two graphs \( G \) and \( H \). In 1994, Livingston and Stout [Constant time computation of minimum dominating sets, Congr. Numer., 105 (1994), 116-128] introduced a linear time algorithm to determine \( \gamma(G \, \Box \, P_n) \) for fixed \( G \), and claimed that \( P_n \) may be substituted with any graph from a one-parameter family, such as a cycle of length \( n \) or a complete \( t \)-ary tree of height \( n \) for fixed \( t \). We explore how the algorithm may be modified to accommodate such graphs and propose a general framework to determine \( \gamma(G \, \Box \, H) \) for any graph \( H \). Furthermore, we illustrate its use in determining the domination number of the generalized Cartesian product \( G \, \Box \, H \), as defined by Benecke and Mynhardt [Domination of Generalized Cartesian Products, preprint (2009)].

We give a solution for the intersection problem for disjoint \( 2 \)-flowers in Steiner triple systems.

Let \( G = (V, E) \) be a graph with chromatic number \( k \). A dominating set \( D \) of \( G \) is called a chromatic-transversal dominating set (ctd-set) if \( D \) intersects every color class of any \( k \)-coloring of \( G \). The minimum cardinality of a ctd-set of \( G \) is called the chromatic transversal domination number of \( G \) and is denoted by \( \gamma_{ct}(G) \). In this paper, we initiate a study of this parameter.

The parity dimension of a graph \( G \) is defined as the dimension of the null space of its closed neighborhood matrix \( N \). A graph with parity dimension \( 0 \) is called all parity realizable (APR). In this paper, a simple recursive procedure for calculating the parity dimension of a tree is given, which is more apt to be used in the context of enumeration than the graph-theoretical characterizations due to Amin, Slater, and Zhang. Applying the recursive relation, we find asymptotic formulas for the number of APR trees and for the average parity dimension of a tree.

The Ramsey multiplicity \( M(G) \) of a graph \( G \) is defined to be the smallest number of monochromatic copies of \( G \) in any two-coloring of edges of \( K_{R(G)} \), where \( R(G) \) is the smallest integer \( n \) such that every graph on \( n \) vertices either contains \( G \) or its complement contains \( G \). With the help of computer algorithms, we obtain the exact values of Ramsey multiplicities for most of isolate-free graphs on five vertices, and establish upper bounds for a few others.