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NOTE ON MINIMUM DEGREE AND PROPER CONNECTION NUMBER

Published online by Cambridge University Press:  03 July 2020

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]

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 $\unicode[STIX]{x1D6FF}(n)>n/36$.

Type
Research Article
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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