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Naduvath, Sudev, Ellumkalayil, Merlin. (1401). A note on δ^(k)-colouring of the Cartesian product of some graphs. , 7(1), 113-120. doi: 10.22049/cco.2021.27114.1211
Sudev Naduvath; Merlin Thomas Ellumkalayil. "A note on δ^(k)-colouring of the Cartesian product of some graphs". , 7, 1, 1401, 113-120. doi: 10.22049/cco.2021.27114.1211
Naduvath, Sudev, Ellumkalayil, Merlin. (1401). 'A note on δ^(k)-colouring of the Cartesian product of some graphs', , 7(1), pp. 113-120. doi: 10.22049/cco.2021.27114.1211
Naduvath, Sudev, Ellumkalayil, Merlin. A note on δ^(k)-colouring of the Cartesian product of some graphs. , 1401; 7(1): 113-120. doi: 10.22049/cco.2021.27114.1211

A note on δ^(k)-colouring of the Cartesian product of some graphs

Communications in Combinatorics and Optimization
مقاله 12، دوره 7، شماره 1، شهریور 2022، صفحه 113-120    اصل مقاله (360.42 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2021.27114.1211
نویسندگان
Sudev Naduvath* 1؛ Merlin Thomas Ellumkalayil2
1Christ University, Bangalore, India.
2Department of Mathematics, Christ University, Bangalore, India.
چکیده
The chromatic number, $\chi(G)$ of a graph $G$ is the minimum number of colours used in a proper colouring of $G$. In an improper colouring, an edge $uv$ is bad if the colours assigned to the end vertices of the edge is the same. Now, if the available colours are less than that of the chromatic number of graph $G$, then colouring the graph with the available colours lead to bad edges in $G$. The number of bad edges resulting from a $\delta^{(k)}$-colouring of $G$ is denoted by $b_{k}(G)$.    In this paper, we use the concept of $\delta^{(k)}$-colouring and determine the number of bad edges in Cartesian product of some graphs.
کلیدواژه‌ها
Improper colouring؛ near proper colouring؛ δ^(k)-colouring؛ bad edge
مراجع
[1] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, New York, 2008.

[2] G. Chartrand and P. Zhang, Chromatic Graph Theory, Chapman and Hall/CRC Press, 2008.

[3] R. Hammack, W. Imrich, and S. Klavžar, Handbook of Product Graphs, CRC Press, 2011.

[4] F. Harary, Graph Theory, Narosa Publ. House, New Delhi., 2001. 

[5] W. Imrich and S. Klavzar, Product Graphs: Structure and Recognition, Wiley, 2000.

[6] T.R. Jensen and B. Toft, Graph Coloring Problems, John Wiley & Sons, 2011.

[7] J. Kok and E.G. Mphako-Banda, Chromatic completion number, J. Math. Comput. Sci. 10 (2020), no. 6, 2971–2983.

[8] J. Kok and S. Naduvath, δ(k)-colouring of cycle related graphs, Adv. Stud. Contemp. Math., to appear.

[9] M. Kubale, Graph Colorings, American Mathematical Soc., 2004.

[10] E.G. Mphako-Banda, An introduction to the k-defect polynomials, Quaest. Math. 42 (2019), no. 2, 207–216.

[11] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, 2001.

 

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