Abstract.

Given an \(n\) vertex graph whose edges have colored from one of \(r\) colors \(C= \{c_1,c_2,\ldots,c_r \}\) , we define the Hamilton cycle color profile \(hcp(G)\) to be the set of vectors \((m_1,m_2,\ldots, m_r)\in [0,n]^r\) such that there exists a Hamilton cycle that is the concatenation of \(r\) paths \(P_1,P_2,\ldots,P_r\) , where \(P_i\) contains \(m_i\) edges of color \(c_i\) . We study \(hcp(G_{n,p})\) when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when \(hcp(G_{n,p})= \{(m_1,m_2,\ldots,m_r)\in [0,n]^r:m_1+m_2+\cdots +m_r=n \}\) .

Keywords

  1. random graph
  2. rainbow Hamilton cycle
  3. path rotation-extention technique

MSC codes

  1. 05C80

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Acknowledgment.

We are grateful to a reviewer for pointing out an error in a previous version and also for an excellent review of our paper.

References

1.
M. Anastos and A. M. Frieze, Pattern colored Hamilton cycles in random graphs, SIAM J. Discrete Math., 33 (2019), pp. 528–545, https://doi.org/10.1137/17M1149420.
2.
D. Bal and A. M. Frieze, Rainbow matchings and Hamilton cycles in random graphs, Random Structures Algorithms, 48 (2016), pp. 503–523.
3.
I. Ben-Eliezer, M. Krivelevich, and B. Sudakov, Long cycles in subgraphs of (pseudo)random directed graphs, J. Graph Theory, 70 (2012), pp. 284–296.
4.
D. Chakraborti and M. Hasabanis, The threshold for the full perfect matching color profile in a random coloring of random graphs, Electron. J. Combin., 28 (2021), 1.21.
5.
C. Cooper and A. M. Frieze, Multi-coloured Hamilton cycles in random edge-coloured graphs, Combin. Probab. Comput., 11 (2002), pp. 129–134.
6.
P. Erdős and A. Rényi, On random matrices, Publ Math. Inst. Hungar. Acad. Sci., 8 (1964), pp. 455–461.
7.
P. Erdős and A. Rényi, On the strength of connectedness of a random graph, Acta Math. Acad. Sci. Hungar., 8 (1961), pp. 261–267.
8.
A. Dudek, A. M. Frieze, and C. Tsourakakis, Rainbow connection of random regular graphs, SIAM J. Discrete Math., 29 (2015), pp. 2255–2266, https://doi.org/10.1137/140998433.
9.
L. Espig, A. M. Frieze, and M. Krivelevich, Elegantly colored paths and cycles in edge colored random graphs, SIAM J. Discrete Math., 32 (2018), pp. 1585–1618, https://doi.org/10.1137/15M1047106.
10.
A. Ferber and M. Krivelevich, Rainbow Hamilton cycles in random graphs and hypergraphs, in Recent Trends in Combinatorics, IMA Vol. Math. Appl. 159, A. Beveridge, J. R. Griggs, L. Hogben, G. Musiker, and P. Tetali, eds., Springer, Cham, 2016, pp. 167–189.
11.
A. M. Frieze, A note on randomly colored matchings in random graphs, in Discrete Mathematics and Applications, Springer Optim. Appl. 165, Springer, Cham, 2020, pp. 199–205.
12.
A. M. Frieze and M. Karoński, Introduction to Random Graphs, Cambridge University Press, 2015.
13.
A. M. Frieze and P. Loh, Rainbow Hamilton cycles in random graphs, Random Structures Algorithms, 44 (2014), pp. 328–354.
14.
A. M. Frieze and C. E. Tsourakakis, Rainbow connectivity of sparse random graphs, Electron. J. Combin., 19 (2012), P5.
15.
A. Heckel and O. Riordan, The hitting time of rainbow connection number two, Electron. J. Combin., 19 (2012), 37.
16.
S. Janson and N. Wormald, Rainbow Hamilton cycles in random regular graphs, Random Structures Algorithms, 30 (2007), pp. 35–49.
17.
N. Kamcev, M. Krivelevich, and B. Sudakov, Some remarks on rainbow connectivity, J. Graph Theory, 83 (2016), pp. 372–383.
18.
C. McDiarmid, Clutter percolation and random graphs, Math. Program. Stud., 13 (1980), pp. 17–25.
19.
M. Molloy, A note on the rainbow connection of random regular graphs, Electron. J. Combin., 24 (2017), P3.49
20.
L. Pósa, Hamilton circuits in random graphs, Discrete Math., 14 (1976), pp. 359–364.

Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 51 - 64
ISSN (online): 1095-7146

History

Submitted: 8 March 2021
Accepted: 27 September 2022
Published online: 20 January 2023

Keywords

  1. random graph
  2. rainbow Hamilton cycle
  3. path rotation-extention technique

MSC codes

  1. 05C80

Authors

Affiliations

Debsoumya Chakraborti Contact the author
Discrete Mathematics Group, Institute for Basic Science IBS, Daejeon 34126, South Korea.
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 USA.
Mihir Hasabnis
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 USA.

Funding Information

Funding: The first author’s research supported by the Institute for Basic Science (IBS-R029-C1). The research of the second author supported in part by NSF grant DMS1952285.

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