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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 [email protected]
SHENWEI HUANG*
Affiliation:
College of Computer Science, Nankai University, Tianjin 300350, China email [email protected]
OWEN MERKEL
Affiliation:
David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1 email [email protected]
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Abstract

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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. 

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.

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