SciELO - Scientific Electronic Library Online

 
vol.38 issue3Stability of two variable pexiderized quadratic functional equation in intuitionistic fuzzy Banach spacesOn a new class of generalized difference sequence spaces of fractional order defined by modulus function author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.3 Antofagasta Aug. 2019

http://dx.doi.org/10.22199/issn.0717-6279-2019-03-0030 

Articles

Rainbow neighbourhood number of graphs

Johan Kok1 

Sudev Naduvath2 

Muhammad Kamran Jamil3 

1CHRIST (Deemed to be University), Department of Mathematics, Bangalore -560029, Karnataka, India. email: kokkiek2@tshwane.gov.za

2CHRIST (Deemed to be University) , Department of Mathematics, Bangalore -560029, Karnataka, India. email: sudev.nk@christuniversity.in

3Riphah International University, Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Lahore, Pakistan. Email : m.kamran.sms@gmail.com

Abstract

In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely the rainbow neighbourhood number and report on preliminary results thereof. The closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one coloured vertex of each colour in the chromatic colouring of a graph is called a rainbow neighbourhood. The number of rainbow neighbourhoods in a graph G is called the rainbow neighbourhood number of G, denoted by rχ(G). We also introduce the concepts of an expanded line graph of a graph G and a v-clique of v ∈ V (G). With the help of these new concepts, we also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph G.

Keywords: Colour cluster; Colour classes; Rainbow neighbourhood; Expanded line graph; v-clique

Mathematics Subject Classification (2000):  05C07; 05C38; 05C75; 05C85

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

References

[1] B. Andrásfai, P. Erdös, and V. Sós, “On the connection between chromatic number, maximal clique and minimal degree of a graph”, Discrete Mathematics, vol. 8, no. 3, pp. 205-218, May 1974, doi: 10.1016/0012-365X(74)90133-2. [ Links ]

[2] J. Bondy and U. Murty, Graph theory, New York: Elsevier Science, 1976. [ Links ]

[3] R. Brooks, “On colouring the nodes of a network”, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 37, no. 2, pp. 194-197, Apr. 1941, doi: 10.1017/S030500410002168X. [ Links ]

[4] G. Chartrand and L. Lesniak, Graphs & digraphs. Boca Raton, FL: Chapman & Hall/CRC, 2000. [ Links ]

[5] G. Chartrand and P. Zhang, Chromatic graph theory. (Discrete Mathematics and Its Applications) Boca Raton, FL: CRC Press, 2009. [ Links ]

[6] R. Green, “Vizing’s theorem and edge-chromatic graph theory”, 2015. [On line]. Available: http://bit.ly/2LZMCwgLinks ]

[7] F. Harary, Graph theory , 5th ed. New Delhi: Narosa Publishing House, 2001. [ Links ]

[8] T. Jensen and B. Toft, Graph coloring problems. New York, NY: John Wiley & Sons, Inc., 1995, doi: 10.1002/9781118032497. [ Links ]

[9] S. Kalayathankal and C. Susanth, “The sum and product of chromatic numbers of graphs and their line graphs”, Journal Of Informatics And Mathematical Sciences, vol. 6, no. 2, pp. 77-85, 2014. [On line]. Available: http://bit.ly/2YF8tLrLinks ]

[10] A. King, B. Reed, and A. Vetta, “An upper bound for the chromatic number of line graphs”, European Journal of Combinatorics, vol. 28, no. 8, pp. 2182-2187, Nov. 2007, doi: 10.1016/j.ejc.2007.04.014. [ Links ]

[11] J. Kok, N. Sudev, and K. Chithra, “Generalised colouring sums of graphs”, Cogent Mathematics, vol. 3, no. 1, Feb. 2016, doi: 10.1080/23311835.2016.1140002. [ Links ]

[12] J. Kok and N. Sudev, “Theb-chromatic number of certain graphs and digraphs”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 19, no. 2, pp. 435-445, Jun. 2016, doi: 10.1080/09720529.2016.1160512. [ Links ]

[13] M. Kubale, Ed., Graph colorings (Contemporary Mathematics, vol. 352). Providence, RI: American Mathematical Society, Jun. 2004, doi: 10.1090/conm/352. [ Links ]

[14] D. West, Introduction to graph theory, 2nd ed. Dehli: Pearson Education Inc. , 2001 [ Links ]

Received: April 2018; Accepted: October 2018

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License