Skip to main content Accessibility help
×
Home

Hamiltonicity in random directed graphs is born resilient

  • Richard Montgomery (a1)

Abstract

Let $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $1/2-\varepsilon$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is ${(1/2-\varepsilon)}$ -resiliently Hamiltonian. Furthermore, for each $\varepsilon > 0$ , we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$ -resiliently Hamiltonian.

This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each $\varepsilon > 0$ , that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$ -resiliently Hamiltonian if $p=\omega(\log^8n/n)$ .

Copyright

Corresponding author

References

Hide All
[1]Ajtai, M., Komlós, J. and Szemerédi, E. (1985) First occurrence of Hamilton cycles in random graphs. North-Holland Math. Studies 115 173178.
[2]Alon, N., Krivelevich, M. and Sudakov, B. (2007) Embedding nearly-spanning bounded degree trees. Combinatorica 27 629644.
[3]Bal, D., Bennett, P., Cooper, C., Frieze, A. and Prałat, P. (2016) Rainbow arborescence in random digraphs. J. Graph Theory 83 251265.
[4]Balogh, J., Csaba, B. and Samotij, W. (2011) Local resilience of almost spanning trees in random graphs. Random Struct. Algorithms 38 121139.
[5]Ben-Shimon, S., Krivelevich, M. and Sudakov, B. (2011) On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs. SIAM J. Discrete Math. 25 11761193.
[6]Bollobás, B. (1983) Almost all regular graphs are Hamiltonian. European J. Combin. 4 97106.
[7]Bollobás, B. (1984) The evolution of sparse graphs. In Graph Theory and Combinatorics: Cambridge Combinatorial Conference in Honour of Paul Erdős, pp. 335357, Academic Press.
[8]Dirac, G. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. 3 6981.
[9]Erdős, P. and Rényi, A. (1959) On random graphs I. Publ. Math. Debrecen 6 290297.
[10]Erdős, P. and Rényi, A. (1964) On random matrices. Magyar Tud. Akad. Mat. Kutató Int. Közl 8 455461.
[11]Ferber, A., Nenadov, R., Noever, A., Peter, U. and Škorić, N. (2017) Robust Hamiltonicity of random directed graphs. J. Combin. Theory Ser. B 126 123.
[12]Frieze, A. (1988) An algorithm for finding Hamilton cycles in random directed graphs. J. Algorithms 9 181204.
[13]Frieze, A. and Krivelevich, M. (2008) On two Hamilton cycle problems in random graphs. Israel J. Math. 166 221234.
[14]Ghouila-Houri, A. (1960) Une condition suffisante d’existence d’un circuit Hamiltonien. CR Acad. Sci. Paris 251 495497.
[15]Glebov, R. (2013) On Hamilton cycles and other spanning structures. PhD thesis.
[16]Hefetz, D., Krivelevich, M. and Szabó, T. (2012) Sharp threshold for the appearance of certain spanning trees in random graphs. Random Struct. Algorithms 41 391412.
[17]Hefetz, D., Steger, A. and Sudakov, B. (2016) Random directed graphs are robustly Hamiltonian. Random Struct. Algorithms 49 345362.
[18]Janson, S., Łuczak, T. and Ruciński, A. (2011) Random Graphs, Wiley.
[19]Komlós, J. and Szemerédi, E. (1983) Limit distribution for the existence of Hamiltonian cycles in a random graph. Discrete Math. 43 5563.
[20]Korshunov, A. (1976) Solution of a problem of Erdős and Rényi on Hamilton cycles in non-oriented graphs. Soviet Math. Dokl. 17 760764.
[21]Krivelevich, M., Lee, C. and Sudakov, B. (2010) Resilient pancyclicity of random and pseudorandom graphs. SIAM J. Discrete Math. 24 116.
[22]Lee, C. and Sudakov, B. (2012) Dirac’s theorem for random graphs. Random Struct. Algorithms 41 293305.
[23]McDiarmid, C. (1983) General first-passage percolation. Adv. Appl. Probab. 15 149161.
[24]Montgomery, R. (2019) Hamiltonicity in random graphs is born resilient. J. Comb. Theory Ser. B. 139 316341.
[25]Nenadov, R., Steger, A. and Trujić, M. Resilience of perfect matchings and Hamiltonicity in random graph processes. Random Struct. Algorithms, 54 797819.
[26]Pittel, B. (1982) On the probable behaviour of some algorithms for finding the stability number of a graph. Math. Proc. Cambridge Philos. Soc. 92 511526.
[27]Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359364.
[28]Rödl, V., Szemerédi, E. and Ruciński, A. (2008) An approximate Dirac-type theorem for k-uniform hypergraphs. Combinatorica 28 229260.
[29]Spencer, J. (1977) Asymptotic lower bounds for Ramsey functions. Discrete Math. 20 6976.
[30]Sudakov, B. (2017) Robustness of graph properties. In Surveys in Combinatorics 2017, Vol. 440 of London Mathematical Society Lecture Note Series, pp. 372408, Cambridge University Press.
[31]Sudakov, B. and Vu, V. (2008) Local resilience of graphs. Random Struct. Algorithms 33 409433.

MSC classification

Hamiltonicity in random directed graphs is born resilient

  • Richard Montgomery (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.