Abstract
Suppose you can colour n biased coins with n colours, all coins having the same bias. It is forbidden to colour both sides of a coin with the same colour, but all other colourings are allowed. Let X be the number of different colours after a toss of the coins. We present a method to obtain an upper bound on a median of X. Our method is based on the analysis of the probability distribution of the number of vertices with even in-degree in graphs whose edges are given random orientations. Our analysis applies to the distribution of the number of vertices with odd degree in random sub-graphs of fixed graphs. It turns out that there are parity restrictions on the random variables that are under consideration. Hence, in order to present our result, we introduce a class of Bernoulli random variables whose total number of successes is of fixed parity and are closely related to Poisson trials conditional on the event that their outcomes have fixed parity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alavi, Y., Erdős, P., Malde, P.J., Schwenk, A.J.: The vertex independence sequence of a graph is not constrained. Congr. Numer. 58, 15–23 (1987)
Alm, S.E., Linusson, S.: A counter-intuitive correlation in a random tournament. Combin. Probab. Comput. 20(1), 1–9 (2011)
Bollobás, B.: Random Graphs, Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge (2001)
Broman, E., van de Brug, T., Kager, W., Meester, R.: Stochastic domination and weak convergence of conditioned Bernoulli random variables. ALEA Lat. Am. J. Probab. Math. Stat. 9(2), 403–434 (2012)
McDiarmir, C.: General percolation and random graphs. Adv. Appl. Probab. 13, 40–60 (1981)
Frank, A., Jordán, T., Szigeti, Z.: An orientation theorem with parity conditions. Discrete Appl. Math. 115, 37–47 (2001)
Hamza, K.: The smallest uniform upper bound on the distance between the mean and the median of the Binomial and Poisson distributions. Stat. Probab. Lett. 23, 21–25 (1995)
Hoeffding, W.: On the Distribution of the Number of Successes in Independent Trials. Ann. Math. Stat. 27, 713–721 (1956)
Kaas, R., Buhrman, J.M.: Mean, median and mode in binomial distributions. Stat. Neerl. 34(1), 13–18 (1980)
Linusson, S.: A note on correlations in randomly oriented graphs (2009, preprint). arXiv:0905.2881
Lovász, L.: Combinatorial Problems and Excercises. North-Holland, Amsterdam (1979)
Pelekis, C., Schauer, M.: Network coloring and colored coin games, chapter 4. In: Alpern, S. et al. (eds.) Search Theory: A Game-Theoretic Prespective (2013)
Shaked, M., Shanthikumar, G.J.: Stochastic Orders and their Applications. Springer, New York (2007)
Tazama, S., Shirakura, T., Tamura, S.: Enumeration of digraphs with given number of vertices of odd out-degree and vertices of odd in-degree. Discrete Math. 90, 63–74 (1991)
Wang, Y.H.: On the number of successes in independent trials. Stat. Sin. 3, 295–312 (1993)
Acknowledgments
I am thankful to Robbert Fokkink, Mike Keane and Tobias Müller for many valuable discussions and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pelekis, C. Bernoulli Trials of Fixed Parity, Random and Randomly Oriented Graphs. Graphs and Combinatorics 32, 1521–1544 (2016). https://doi.org/10.1007/s00373-015-1650-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-015-1650-2