**Authors:** Paul August Winter

Much research has been done involving the chromatic number of a graph involving the least number of colors, that the vertices of a graph can be colored, so that no two adjacent vertices have the same color. The idea of how the chromatic number of a vertex cover of a graph dominates the vertex cover of the original graph, where a large number of vertices are involved, has been investigated. The difference between the energy of the complete graph,, and the energy of any other graph G. has been studied, in terms of a ratio. The complete graph, on n vertices, has chromatic number n, and is significant in terms of its easily accessible graph theoretical properties, such as its high level of connectivity and robustness. In this paper, we introduce a ratio, the chromatic-complete difference ratio, involving the difference between the chromatic number of the complete graph, and the chromatic number of any other connected graph G, on the same number n of vertices. This allowed for the investigation of the effect of the chromatic number of G, with respect to the complete graph, when a large number of vertices are involved - referred to as the chromatic-complete difference domination effect. The value of this domination effect lies on the interval [0,1], with most classes of graphs taking on the right hand end-point, while graphs with a large clique takes on the left hand end-point. When this ratio is a function f(n), of the order of a graph, we attach the average degree of G to the Riemann integral to investigate the chromatic-complete difference area aspect of classes of graphs. We applied these chromatic-complete difference aspects to complements of classes of graphs. AMS Classification: 05C50 1Corresponding author: Paul August Winter: Department of Mathematics, Howard College, University of KwaZulu-Natal, Glenwood, Durban, 4041, South Africa; ORCID ID: 0000-0003-3539; email: winterp@ukzn.ac.za Key words: Chromatic number, domination, ratios, domination, asymptotes, areas

**Comments:** 23 Pages.

**Download:** **PDF**

[v1] 2015-03-07 06:42:57

**Unique-IP document downloads:** 59 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful. *