Abstract
The burning number b(G) of a graph G was introduced by Bonato, Janssen, and Roshanbin [Lecture Notes in Computer Science 8882 (2014)] to measure the speed of the spread of contagion in a graph. They proved for any connected graph G of order n, \(b(G)\le 2\lceil \sqrt{n} \rceil -1\), and conjectured that \(b(G)\le \lceil \sqrt{n} \rceil \). In this paper, we proved \(b(G)\le \lceil \frac{-3+\sqrt{24n+33}}{4}\rceil \), which is roughly \(\frac{\sqrt{6}}{2}\sqrt{n}\). We also settled the following conjecture of Bonato-Janssen-Roshanbin: \(b(G)b(\bar{G})\le n+4\) provided both G and \(\bar{G}\) are connected.
L. Lu—This author was supported in part by NSF grant DMS-1600811.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Biggs, N.L., Lloyd, E.K., Wilson, R.J.: Graph Theory 1736–1936. Oxford University Press, Oxford (1976)
Bonato, A., Janssen, J., Roshanbin, E.: Burning a graph as a model of social contagion. In: Bonato, A., Graham, F.C., Prałat, P. (eds.) WAW 2014. LNCS, vol. 8882, pp. 13–22. Springer, Heidelberg (2014). doi:10.1007/978-3-319-13123-8_2
Bonato, A., Janssen, J., Roshanbin, E.: How to burn a graph. Internet Math. 12(1–2), 85–100 (2016)
Banerjee, S., Das, A., Gopalan, A., Shakkottai, S.: Epidemic spreading with external agents. IEEE Trans. Inf. Theor. 60(7), 4125–4138 (2014)
Bessy, S., Bonato, A., Janssen, J., Rautenbach, D., Roshanbin, E.: Bounds on the Burning Number. Submitted to Discrete Applied Mathematics
Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discret. Math. 307, 2094–2105 (2007)
Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Australas. J. Comb. 43, 57–77 (2009)
Mitsche, D., Pralat, P., Roshanbin, E.: Burning number of graph products. Submitted to Theoretical Computer Science
Mitsche, D., Pralat, P., Roshanbin, E.: Burning graphs—a probabilistic perspective. Submitted to Graphs and Combinatorics
Roshanbin, E.: Burning a graph as a model of social contagion. Ph.D. thesis, Dalhousie University (2016)
Acknowledgment
We would like to thank the anonymous referees for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Land, M.R., Lu, L. (2016). An Upper Bound on the Burning Number of Graphs. In: Bonato, A., Graham, F., Prałat, P. (eds) Algorithms and Models for the Web Graph. WAW 2016. Lecture Notes in Computer Science(), vol 10088. Springer, Cham. https://doi.org/10.1007/978-3-319-49787-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-49787-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49786-0
Online ISBN: 978-3-319-49787-7
eBook Packages: Computer ScienceComputer Science (R0)