A defective vertex coloring of a graph is a coloring in which some adjacent vertices may have the same color. An edge whose adjacent vertices have the same color is called a bad edge. A defective coloring of a graph $G$ with minimum possible number of bad edges in $G$ is known as a near proper coloring of $G$.  In this paper, we introduce the notion of equitable near proper coloring of graphs and determine the minimum number of bad edges obtained from an equitable near proper coloring of some graph classes.

In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and the icosahedral graph.

The perfect m-coloring with matrix A = [aij ]i, j2f1,
,
,
, ,mg of a graph G = (V, E) with f1,
,
,
, ,mg color is a vertex coloring of G with m-color so that the number of vertices in color j adjacent to a , xed vertex in color i is aij, independent of the choice of vertex in color i. The matrix A = [aij ]i, j2f1,
,
,
, ,mg is called the parameter matrix. We study the perfect 4-colorings of the 3-regular graphs of order at most 8, that is, we determine a list of all color parameter matrices corresponding to perfect 4-colorings of 3-regular graphs of orders 4, 6, and 8.

A dominating set $S$ of $G$ is an \textit{equitable dominating set} of $G$ if for every $v \in V(G) \setminus S$, there exists $u \in S$ such that $uv \in V(G)$ and $\displaystyle{\left|\deg(u) - \deg(v)\right| \leq 1.}$ A dominating set $S$ of $G$ is a \textit{rings dominating set} of $G$ if every vertex $v \in V(G) \setminus S$ is adjacent to atleast two vertices $V(G) \setminus S$. In this paper, we examine the conditions at which the equitable dominating set and the rings dominating set coincide, and thus naming the dominating set as \textit{equitable rings dominating set}. The minimum cardinality of an equitable rings dominating set of a graph $G$ is called the \textit{equitable rings domination number} of $G$, and is denoted by $\gamma_{eri}(G)$. Moreover, we examine determine the equitable rings domination number of many graphs, and graphs formed by some binary operations.

A dominating set $S$ of $G$ is an \textit{equitable dominating set} of $G$ if for every $v \in V(G) \setminus S$, there exists $u \in S$ such that $uv \in V(G)$ and $\displaystyle{\left|\deg(u) - \deg(v)\right| \leq 1.}$ A dominating set $S$ of $G$ is a \textit{rings dominating set} of $G$ if every vertex $v \in V(G) \setminus S$ is adjacent to atleast two vertices $V(G) \setminus S$. In this paper, we examine the conditions at which the equitable dominating set and the rings dominating set coincide, and thus naming the dominating set as \textit{equitable rings dominating set}. The minimum cardinality of an equitable rings dominating set of a graph $G$ is called the \textit{equitable rings domination number} of $G$, and is denoted by $\gamma_{eri}(G)$. Moreover, we examine determine the equitable rings domination number of many graphs, and graphs formed by some binary operations.

Freshwater lens on small islands may easily be overexploited or polluted due to overdrafts of fresh water by pumping which causes drawdown of the water table a rise or up-coning of the saltwater interface. Present study is concerned with using a three-dimensional finite-difference numerical model to simulate the groundwater flow and transport model to predict the behavior of groundwater system in Manukan Island. The simulations were done using variable density SEAWAT-2000 for three selected pumping schemes. Hydraulic heads (groundwater level) is the highest at the center of the island and decreases in radial shape towards the coast in all the pumping schemes (PS1-PS3). The chloride concentration in the studied aquifer increased by 98.7% in the pumping well if the pumping rate is doubled by the current (PS2 to PS3). The 1.4% seawater-freshwater mixing moves further forward to inland about 1.6m when the current pumping rate was doubled whereas moves backward to sea about 1.7m if the current pumping rate is reduced by 50%. This preliminary model of Manukan island aquifer shows that an overexploitation of groundwater in Manukan Island contributes to the seawater intrusion. Adjusting the future groundwater pumping scheme and improving groundwater management strategies are necessary to protect the freshwater aquifers. The current numerical model is a reasonable representation of the aquifer in Small Island which can be used in similar small islands with similar hydrogeological conditions in elsewhere.

In this paper, we study perfect 2-coloring of the quartic graphs with at most 8 vertices. The problem of the existence of perfect coloring is a generalization of the concept of completely regular codes, given by Delsarte.

We study the problem of online coloring co-interval graphs. In this problem, a set of intervals on the real line is presented to the algorithm, one at a time, and upon receiving each interval I, the algorithm must assign I a color different from the colors of all previously presented intervals not intersecting I. The objective is to use as few colors as possible. It is known that the competitive ratio of the simple FIRST-FIT algorithm on the class of co-interval graphs is at most 2. We show that for the class of unit co-interval graphs, where all intervals have equal length, the 2-bound on the competitive ratio of FIRST-FIT is tight. On the other hand, we show that no deterministic online algorithm for coloring unit co-interval graphs can be better than 3/2-competitive. We then study the effect of randomization on our problem and show a lower bound of 4/3 on the competitive ratio of any randomized algorithm for the unit co-interval coloring problem. We also prove that for the class of general co-interval graphs, no randomized algorithm has a competitive ratio better than 3/2.

An equitable domination has interesting application in the context of social networks. In a network, nodes with nearly equal capacity may interact with each other in a better way. In the society persons with nearly equal status, tend to be friendly. In this paper, we introduce new variant of equitable domination of a graph. Basic properties and some interesting results have been obtained.