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Automated discovery and proof in three combinatorial problems
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Automated discovery and proof in three combinatorial problems
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ETD_2062
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000051892
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eng
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theses
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Mathematics
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Combinatorial analysis
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Graph theory
Abstract
In this Ph.D. disseration, I will go over advances I have made in three combinatorial problems. The running theme throughout these three problems is the novel use of computers to aid not only in the discovery of the theorems proved, but also in the proofs themselves. The first problem concerns the quantity f_D(n), defined as the size of the largest subset of [n] avoiding differences in D. Originally motivated by the Triangle Conjecture of Schutzenberger and Perrin, we again define an enumeration scheme that will find, and prove automatically, the sequence {f_D(n)} from 1 to infinity for any prescribed D. Although the Triangle Conjecture has long been refuted, we present an asymptotic version of it and prove it. The second problem involves the enumeration of spanning trees in grid graphs – graphs of the form G x P_n (or C_n) for arbitrary G. An enumeration scheme is developed based on the partitions of [n], yielding an algorithmic method to completely solve the sequence for any G. These techniques yield a surprising consequence: sequences obtained in this manner are divisibility sequences. The
final problem is the firefighter problem, a dynamic graph theory problem modeling the spread of diseases, information, etc. We will present the problem as it applies on the two-dimensional grid and prove new upper and lower bounds, found mainly through computer experimentation.
final problem is the firefighter problem, a dynamic graph theory problem modeling the spread of diseases, information, etc. We will present the problem as it applies on the two-dimensional grid and prove new upper and lower bounds, found mainly through computer experimentation.
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electronic resource
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x, 118 p. : ill.
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Ph.D.
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Includes bibliographical references (p. 114-117)
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by Paul Raff
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Raff
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Paul
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1982-
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Paul Raff
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Zeilberger
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Doron
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chair
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Doron Zeilberger
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Roberts
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Fred
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Fred Roberts
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Retakh
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Vladimir
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Vladimir Retakh
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Sloane
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Neil
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Neil Sloane
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Rutgers University
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Graduate School - New Brunswick
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2009
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2009-10
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xx
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Rutgers University Electronic Theses and Dissertations
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ETD
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T3T72HM0
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ETD doctoral
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The author owns the copyright to this work
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Copyright protected
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Open
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Raff
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Paul
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Paul Raff
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Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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