On equitable coloring of book graph families

Abstract

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by atmost one. The notion of equitable coloring was introduced by Meyer in 1973. A proper $h-$colorable graph $K$ is said to be equitably h-colorable if the vertex sets of $K$ can be partioned into $h$ independent color classes $V_1, V_2,...,V_h$ such that the condition $\left|\left|V_i\right|-\left|V_j\right|\right| \leq 1$ holds for all different pairs of $i$ and $j$ and the least integer $h$ is known as equitable chromatic number of $K$. In this paper, we find the equitable coloring of book graph, middle, line and central graphs of book graph.

Keywords

• equitable coloring
• book graph
• middle graph
• line graph
• central graph

References

1. Furmanczyk, H., Equitable coloring of Graph products, Opuscula Mathematica, Vol 26. No.1, (2006).
2. Harary, F., Graph theory, Narosa Publishing home, New Delhi, 1969.
3. Hajnal, A., Szemeredi, E., Proof of a conjecture of Endos, in: Combinatorial theory and its applications, Colloq. Math. Soc. Janos Bolyai, 4 (2) (1970), 601-623.
4. Meyer, W., Equitable coloring, Amer. Math. Monthly, 80, (1973).
5. Michalak, D., On middle and total graphs with coarseness number equal 1, Springer Verlag Graph Theory, Lagow Proceedings, Berlin Heidelberg, New York, Tokyo, (1981), 139-150.
6. Vernold Vivin, J., Harmonious coloring of total graphs, n-leaf, central graphs and circumdetic graphs, Bharathiar University, Ph.D Thesis, Coimbatore, India, 2007.