Study on Eccentric Coloring of a Graph

Abstract

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). An eccentric coloring of a graph G=(V,E) is a function color: V\(\to\)N such that

(i) for all u,v \(\epsilon\) V,(color(u)=color(v)) \(\Rightarrow\) d(u,v)>color(u).
(ii) for all v \(\epsilon\) V,color(v) \(\le\) e(v).

The eccentric chromatic number X_e \(\epsilon\) N for a graph G is the lowest number of colors for which it is possible to eccentrically color G by colors: V \(\to\) {1,2,…,X_e }. In this paper, we have considered eccentric colorability of a graph in relation to other properties. we have considered simple undirected graphs without multiple edges and self loops. Also, we have considered the eccentric colorability of lexicographic product of some special class of graphs.