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A BOUND FOR THE CHROMATIC NUMBER OF ( $P_{5}$ , GEM)-FREE GRAPHS

Published online by Cambridge University Press:  28 March 2019

KATHIE CAMERON
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 email kcameron@wlu.ca
SHENWEI HUANG
Affiliation:
College of Computer Science, Nankai University, Tianjin 300350, China email dynamichuang@gmail.com
OWEN MERKEL
Affiliation:
David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1 email owen.merkel@uwaterloo.ca

Abstract

As usual, $P_{n}$ ( $n\geq 1$ ) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$ . A graph is called ( $P_{5}$ , gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$ , $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$ . We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ( $P_{5}$ , gem)-free graph $G$ .

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

Shenwei Huang is the corresponding author. The research of the first author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06517; the research of the second author was supported by the National Natural Science Foundation of China grant 11801284; the research of the third author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06517 and an NSERC Undergraduate Student Research Award.

References

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