Graphs and Combinatorics

, Volume 34, Issue 4, pp 677–692 | Cite as

On the Chromatic Number of (\(P_6\), Diamond)-Free Graphs

  • T. Karthick
  • Suchismita Mishra
Original Paper


In this paper, we first show that every (\(P_6\), diamond, \(K_4\))-free graph is 6-colorable. We also give an example of a (\(P_6\), diamond, \(K_4\))-free graph G with \(\chi (G)\) \( = 6\). Further, we show that for every (\(P_6\), diamond)-free graph G, the chromatic number of G is upper bounded by a linear function of its clique number. This generalizes some known results in the literature.


Graph classes \(P_6\)-free graphs Diamond-free graphs Chromatic number Clique number 



The first author thanks Mathew C.Francis for his comments. The authors are grateful to the anonymous referees for their helpful suggestions and remarks.


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© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Computer Science UnitIndian Statistical Institute, Chennai CentreChennaiIndia
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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