In this paper, we study Perfect 2-Coloring of the six-regular graphs up to order 10. We first obtain all possible color parameter matrices and then examine them according to the limitation of our theorem.

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.

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.

Let‎ ‎$G =(V_G‎ , ‎E_G)$‎ ‎be a graph and let $I$ be a finite set of size $m\geq 1$‎. ‎A mapping $T:V_G \rightarrow I$‎ ‎is called a Perfect $m$-Coloring with a parameter matrix $A = (a_{ij})_{i,j\in I }$‎ ‎of $G$ if it is surjective and for all $i,j$‎, ‎every vertex of color $i$ has $a_{i j}$ neighbors of color $j$‎. ‎In this paper‎, ‎we classify all the realizable parameter matrices of Perfect 3-Colorings of the line graphs of the connected bicubic graphs of order at most 12‎.

The Perfect m-Coloring with matrix A = [aij ]i, j2f1,
,
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, ,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.

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.

A rapidly developing field of science and technology is nanobiotechnology. Nanotube, nanostar and polyomino chain are critical and widespread molecular structures extensively used in the domains of pharmaceuticals, chemical engineering, and medical science. Additionally, these structures serve as the foundational building blocks for other, more intricate chemical molecular structures. In this paper, certain chemical structures like nanostar dendrimer, oxide network, silicate network, boron nanosheet and polyomino chains have been acyclically colored using the concept of vertex cut and matching. Also, we determine the acyclic Coloring parameters for the networks under consideration and find a relation between them.