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NOTE ON MINIMUM DEGREE AND PROPER CONNECTION NUMBER

Part of: Graph theory

Published online by Cambridge University Press:  03 July 2020

YUEYU WU Open the ORCID record for YUEYU WU [Opens in a new window] ,
YUNQING ZHANG Open the ORCID record for YUNQING ZHANG [Opens in a new window]  and
YAOJUN CHEN Open the ORCID record for YAOJUN CHEN [Opens in a new window]
YUEYU WU
Affiliation:
Department of Mathematics, Nanjing University, Nanjing210093, PR China email [email protected]
YUNQING ZHANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing210093, PR China email [email protected]
YAOJUN CHEN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing210093, PR China email [email protected]
*
✉[email protected]
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Abstract

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$, of a graph $G$, is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$. Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)>n/42$. In this note, we show that $\unicode[STIX]{x1D6FF}(n)>n/36$.

Keywords

proper colouringproper connection numberminimum degree

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C07: Vertex degrees 05C40: Connectivity
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 177 - 181
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by NSFC under Grant Nos 11671198, 11871270 and 11931006.

References

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Brause, C., Doan, T. and Schiermeyer, I., ‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin. 33 (2017), 833–843.Google Scholar
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