Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 4 (2019), 659-670.

Graphs with at most two trees in a forest-building process

Steve Butler, Misa Hamanaka, and Marie Hardt

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Abstract

Given a graph, we can form a spanning forest by first sorting the edges in a random order, and then only keeping edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with k trees.

We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are nonisomorphic but produce the same probabilities.

Article information

Source
Involve, Volume 12, Number 4 (2019), 659-670.

Dates
Received: 30 March 2018
Revised: 10 September 2018
Accepted: 28 October 2018
First available in Project Euclid: 30 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1559181658

Digital Object Identifier
doi:10.2140/involve.2019.12.659

Mathematical Reviews number (MathSciNet)
MR3941604

Zentralblatt MATH identifier
07072545

Subjects
Primary: 05C05: Trees

Keywords
forests edge ordering components probability

Citation

Butler, Steve; Hamanaka, Misa; Hardt, Marie. Graphs with at most two trees in a forest-building process. Involve 12 (2019), no. 4, 659--670. doi:10.2140/involve.2019.12.659. https://projecteuclid.org/euclid.involve/1559181658


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References

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  • S. Butler, F. Chung, J. Cummings, and R. Graham, “Edge flipping in the complete graph”, Adv. in Appl. Math. 69 (2015), 46–64.
  • R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics: a foundation for computer science, 2nd ed., Addison-Wesley, Reading, MA, 1994.