More results on eccentric coloring in graphs

Medha Itagi Huilgol

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.


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How to Cite this Article:

Medha Itagi Huilgol, More results on eccentric coloring in graphs, Journal of Mathematical and Computational Science, Vol 9, No 2 (2019), 225-238

Copyright © 2019 Medha Itagi Huilgol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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