Abstract
Given a finite word w over a finite alphabet V, we may construct a graph with vertex set V and an edge between to elements of V if and only if they alternate in the word w. This is the notion of word-representability of graphs. In this paper, we first study minimal length words which represent graphs, giving an explicit formula for both the length and the number of such words in the case of trees and cycles. Then we extend this notion to study the graphs representable with other patterns in words, proving in all cases aside from one (still unknown to us), all graphs are representable by all other patterns. Finally, we pose a few open problems for further work.
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Acknowledgments
This research was conducted at the University of Minnesota Duluth REU, funded by NSF Grant 1650947 and NSA Grant H98230-18-1-0010. The authors would like to thank Joe Gallian for running the REU and suggesting the topic. The authors also thank their advisors Levent Alpoge, Aaron Berger, and Colin Defant for their support, as well as the anonymous reviewers for many helpful comments.
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Gaetz, M., Ji, C. (2019). Enumeration and Extensions of Word-Representants. In: Mercaş, R., Reidenbach, D. (eds) Combinatorics on Words. WORDS 2019. Lecture Notes in Computer Science(), vol 11682. Springer, Cham. https://doi.org/10.1007/978-3-030-28796-2_14
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DOI: https://doi.org/10.1007/978-3-030-28796-2_14
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