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On Coloring Catalan Number Distance Graphs and Interference Graphs

1
Department of Mathematics, SASTRA Deemed to be University, Thanjavur 613401, Tamilnadu, India
2
Department of Electronics and Communication Engineering, SSN College of Engineering, Chennai 603110, Tamilnadu, India
3
Department of Automation, Industrial Engineering, Textiles and Transport, University ‘Aurel Vlaicu’, B-dul Revolutiei nr. 77, 310130 Arad, Romania
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2018, 10(10), 468; https://doi.org/10.3390/sym10100468
Received: 29 August 2018 / Revised: 2 October 2018 / Accepted: 5 October 2018 / Published: 9 October 2018
A vertex coloring of a graph G is a mapping that allots colors to the vertices of G. Such a coloring is said to be a proper vertex coloring if two vertices joined by an edge receive different colors. The chromatic number χ ( G ) is the least number of colors used in a proper vertex coloring. In this paper, we compute the χ of certain distance graphs whose distance set elements are (a) a finite set of Catalan numbers, (b) a finite set of generalized Catalan numbers, (c) a finite set of Hankel transform of a transformed sequence of Catalan numbers. Then while discussing the importance of minimizing interference in wireless networks, we probe how a vertex coloring problem is related to minimizing vertex collisions and signal clashes of the associated interference graph. Then when investigating the χ of certain G ( V , D ) and graphs with interference, we also compute certain lower and upper bound for χ of any given simple graph in terms of the average degree and Laplacian operator. Besides obtaining some interesting results we also raised some open problems. View Full-Text
Keywords: chromatic number; distance graphs; interference graphs; Catalan numbers; Hankel transforms; chromatic threshold chromatic number; distance graphs; interference graphs; Catalan numbers; Hankel transforms; chromatic threshold
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MDPI and ACS Style

Yegnanarayanan, V.; Yegnanarayanan, G.N.; M. Balas, M. On Coloring Catalan Number Distance Graphs and Interference Graphs. Symmetry 2018, 10, 468. https://doi.org/10.3390/sym10100468

AMA Style

Yegnanarayanan V, Yegnanarayanan GN, M. Balas M. On Coloring Catalan Number Distance Graphs and Interference Graphs. Symmetry. 2018; 10(10):468. https://doi.org/10.3390/sym10100468

Chicago/Turabian Style

Yegnanarayanan, Venkataraman, Gayathri N. Yegnanarayanan, and Marius M. Balas. 2018. "On Coloring Catalan Number Distance Graphs and Interference Graphs" Symmetry 10, no. 10: 468. https://doi.org/10.3390/sym10100468

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