Department of Mathematics, Howard College, University of KwaZulu-Natal, Glenwood, Durban, 4041, South Africa.
Chromatic number ,Vertex cover ,Domination ,Ratios ,Asymptotes ,Areas , Farey sequences ,

The study of the chromatic number (the smallest number of colors required to color the vertices of a graph so that adjacent vertices have different colors) and vertex covering (the smallest set S of vertices so that every edge of the graph has at least one end in S) of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, to investigate the domination effect of the chromatic number, of the subgraph induced by a vertex covering of a graph G, on the original chromatic number of G, where large number of vertices is involved. This is referred to as the chromatic-cover domination. The chromatic idea can be translated to a molecular construction where atoms ae partitioned into the smallest number of color classes (atom classes), where same color atoms are not bonded and only different colored atoms are bonded. The covering can be translated to a molecular structure where a large number of atoms are involved, and the smallest set S of atoms are needed, in terms of conservation of energy, where all bonds (edges) between atoms (vertices) have at least one atom in S, so that atomic activation of S would activate all atoms through the bonds. If this chromatic-cover ratio is a function of n, the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating chromatic-cover area with classes of graphs. We found that the chromatic-cover domination had a strongest effect on complete graphs, while this chromatic-cover domination had zero effect on star graphs. We show that the chromatic-cover asymptote of all classes of graphs belong to the interval [0,1]. We construct a class of graphs, using known classes of graph, where end vertices are replaced with cliques on q vertices, thus generating sequences. We use a particular sequence to construct a sequence which is a subsequence of the famous Farey sequence.

5 , 12 , 2015