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.

The ATSP polytope can be expressed by an asymmetric polynomial-size linear program.

A model that represents the rate of changes of the population with limited environmental resources can be described by,

\[
\frac{dp}{dt} = p\left(a – {bp}\right) + g(t,p) = p(t_0)= p_0
\]

where \( a \) measures the growth rate in the absence of the restriction force \( b \) and \( \frac{a}{b} \) is called the carrying capacity of the environment. The random perturbation \( g(t,P) \) is generated by random change in the environment. The behavior of the solution of this model for continuous and discrete case when \( g(t,P)=w(t) \) is density independent with a constant random factor \( w \) in a short time interval \([t, t + \delta t)\) will be studied. The stability and the behavior of the equilibrium point will also be investigated. A computational approach to the solution using Excel spreadsheet and Maple will be presented.

For a set \( S \) of two or more vertices in a nontrivial connected graph \( G \) of order \( n \), a collection \(\{T_1, T_2, \ldots, T_\ell\}\) of trees in \( G \) is said to be an internally disjoint set of trees connecting \( S \) if these trees are pairwise edge-disjoint and \( V(T_i) \cap V(T_j) = S \) for every pair \( i, j \) of distinct integers with \( 1 \leq i, j \leq \ell \). For an integer \( k \) with \( 2 \leq k \leq n \), the tree \( k \)-connectivity \( \kappa_k(G) \) of \( G \) is the greatest positive integer \( \ell \) for which \( G \) contains at least \( \ell \) internally disjoint trees connecting \( S \) for every set \( S \) of \( k \) vertices of \( G \). It is shown for every two integers \( k \) and \( r \) with \( 3 \leq k \leq 2r \) that
\[
\kappa_k(K_{r,r}) = r – \left\lceil \frac{k-1}{4} \right\rceil.
\]

This paper investigates the existence of monadic balanced ternary designs (BTDs). A monadic BTD is a BTD where each size \( K \) block contains one element that appears doubly and \( K-2 \) elements that appear singly. The authors show that the conditions

1. \( \rho_1 = 2\rho_2 \),
2. \( \Lambda(V-1) = 10\rho_2 \),
3. \( \Lambda \neq 3 \),

are sufficient for the existence of monadic BTDs \( (V; B; \rho_1, \rho_2, R; 4; \Lambda) \). The authors also give necessary and sufficient conditions for the existence of monadic BTDs where the block size is five and \( \Lambda \) is 3 or 6.

We consider the placement of detection devices at the vertices of a graph \( G \), where a detection device at vertex \( v \) has three possible outputs: there is an intruder at \( v \); there is an intruder at one of the vertices in the open neighborhood \( N(v) \), the set of vertices adjacent to \( v \), but which vertex in \( N(v) \) cannot be determined; or there is no intruder in \( N[v] = N(v) \cup \{v\} \). We introduce the \( 1 \)-step locating-dominating problem of placing the minimum possible number of such detection devices in \( V(G) \) so that the presence of an intruder in \( V(G) \) can be detected, and the exact location of the intruder can be identified, either immediately or when the intruder has moved to an adjacent vertex. Some related problems are introduced.

Recently, four new vertex colorings of graphs (in which adjacent vertices may be colored the same) were introduced for the purpose of distinguishing every pair of adjacent vertices. For each graph and for each of these four colorings, the minimum number of required colors never exceeds the chromatic number of the graph. In this paper, we summarize some of the results obtained on these colorings and introduce some relationships among them.

We address the problem: for which values of \( d \) and \( n \) does there exist a triangle-free regular graph of degree \( d \) on \( n \) vertices? A complete solution is given.