Coloring sums of extensions of certain graphs

Year 2018, , 19 - 27, 15.01.2018

Johan Kok Saptarshi Bej

https://doi.org/10.13069/jacodesmath.349383

Abstract

We recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted $\chi(G)$. Motivated by the introduction of the concept of the $b$-chromatic sum of a graph the concept of $\chi'$-chromatic sum and $\chi^+$-chromatic sum are introduced in this paper. The extended graph $G^x$ of a graph $G$ was recently introduced for certain regular graphs. This paper furthers the concepts of $\chi'$-chromatic sum and $\chi^+$-chromatic sum to extended paths and cycles. Bipartite graphs also receive some attention. The paper concludes with patterned structured graphs. These last said graphs are typically found in chemical and biological structures.

Keywords

Chromatic number, $\chi'$-chromatic sum, $\chi^+$-chromatic sum, Extended path, Extended cycle

References

• [1] A. Banerjee, S. Bej, On extension of regular graphs, arXiv:1509.05476v1 [math.CO].
• [2] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, 2008.
• [3] S. Cabello, M. Jacovac, On the b–chromatic number of regular graphs, Discrete Appl. Math. 159 (2011) 1303–1310.
• [4] G. Chartrand, L. Lesniak, Graphs and Digraphs, CRC Press, 2000.
• [5] J. T. Gross, J. Yellen, Graph Theory and Its Applications, CRC Press, 2006.
• [6] J. E. Hopcroft, R. M. Karp, An $n^{5/2}$ algorithm for maximum matchings in bipartite graphs, SIAM J. Comput. 2(4) (1973) 225–231.
• [7] J. Kok, N. K. Sudev, K. P. Chithra, Generalised colouring sums of graphs, Cogent Math. 3(1) (2016) 1–11.
• [8] M. Kouider, A. El Sahili, About b–coloring of regular graphs, Rapport de Recherche, No. 1432, CNRS–Universite Paris Sud–LRI.
• [9] E. Kubicka, A. J. Schwenk, An introduction to chromatic sums, Proc. ACM Computer Sci. Conf. (Louisville) (1989) 39–45.
• [10] P. C. Lisna, M. S. Sunitha, b–chromatic sum of a graph, Discrete Math. Algorithm. Appl. 7(4) (2015) 1–15.
• [11] N. K. Sudev, K. P. Chithra, J. Kok, Certain chromatic sums of some cycle-related graph classes, Discrete Math. Algorithm. Appl. 8(3) (2016) 1–25.