More results on eccentric coloring in graphs

Abstract

The eccentricity $ e(u) $ of a vertex $ u $ is the maximum distance from $u$ in $G$. A vertex $v$ is an eccentric vertex of $ 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: $V \longrightarrow N$ such that

(i)  for all $u, v \in V$, $ (color(u) = color(v))\Longrightarrow d(u,v) > color(u) $,

(ii)  for all $v \in V$, color $ (v)\leq e(v)$.

The eccentric chromatic number $ \chi_{e} \in N $ for a graph $ G $ is the least number of colors for which it is possible to eccentrically color $ G $ by colors: $V \longrightarrow \lbrace1,2,....,\chi_{e}\rbrace. $ In this paper, we have proved that a cycle with a chord between vertices at any distance up to the radius of the cycle is eccentric colorable thereby making the results of [5] particular cases. Also, here we have extended results on eccentric coloring of Lexicographic product graphs proved earlier and found a sharp upper bound and shown its attainability.