Abstract
A vertex coloring of a graph G is a mapping that allots colors to the vertices of G. Such a coloring is said to be a proper vertex coloring if two vertices joined by an edge receive different colors. The chromatic number χ ( G ) is the least number of colors used in a proper vertex coloring. In this paper, we compute the χ of certain distance graphs whose distance set elements are (a) a finite set of Catalan numbers, (b) a finite set of generalized Catalan numbers, (c) a finite set of Hankel transform of a transformed sequence of Catalan numbers. Then while discussing the importance of minimizing interference in wireless networks, we probe how a vertex coloring problem is related to minimizing vertex collisions and signal clashes of the associated interference graph. Then when investigating the χ of certain G ( V , D ) and graphs with interference, we also compute certain lower and upper bound for χ of any given simple graph in terms of the average degree and Laplacian operator. Besides obtaining some interesting results we also raised some open problems.
Highlights
A graph comprising vertices and edges is a discrete structure in which each edge joins only two different vertices
The chromatic number χ( G ) is the least number of colors used in a proper vertex coloring
While discussing the importance of minimizing interference in wireless networks, we probe how a vertex coloring problem is related to minimizing vertex collisions and signal clashes of the associated interference graph
Summary
A graph comprising vertices and edges is a discrete structure in which each edge joins only two different vertices. To find the number of frequencies required one has to find the minimum cardinality of the lists that allows the vertices of the graph colored without violating proper coloring concept This number is named as the list chromatic number and its computation is much more hard to find than the usual χ. Several variables can be allotted to a given register, but variables that are in use at a given time cannot be allotted to the same register without spoiling their values When we model this by means of a graph, vertices stand for temporary variables and two of them are joined by an edge if they are involved concurrently at some point in the program. The number of registers required to make the program run is equal to the least number of colors in a proper vertex coloring of this interference graph. For higher dimensions of the form R × R × · · · × R n-times the lower and upper bound for the χ( G ( R × R × · · · × R, {1})) are (1 + o (1)).1.2d and (3 + o (1))d in [14] and [15] respectively
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