Oscar Levin

Faculty

Professor

Email oscar.levin@unco.edu
Phone (970) 351-2380
Location Ross Hall
Address 501 20th St, Campus Box 122, Greeley, CO 80639

Education

PhD, University of Connecticut, 2009
Area of Study: Mathematics

MS, University of Connecticut, 2006
Area of Study: Mathematics

BA, University of Northern Colorado, 2004
Area of Study: Philosophy

BS, University of Northern Colorado, 2004
Area of Study: Mathematics

Professional Experience & Affiliations

Professor, University of Northern Colorado
School of Mathematical Sciences (2023 – Present)

Associate Professor, University of Northern Colorado
(2017-present)

Assistant Professor, University of Northern Colorado
(2011-2017)

Visiting Assistant Professor, Coastal Carolina University
(2009-2011)

Graduate Teaching Assistant, University of Connecticut
(2005-2009)

Research Expertise & Interests

Generic and equitable colorings in computable graphs

An equitable coloring of a graph is a partition of its vertices into equal-sized sets so that adjacent vertices belong to different sets. We explore the extent to which nicely described (computable) graphs might have equitable colorings that are not nicely described. This sheds light on the limits of computational ability for graphs.

Areas of Interest

Courses I regularly teach include Discrete Mathematics (MATH 228), and Abstract Algebra I & II (MATH 321 & 322). I have also taught courses on Calculus, Modern Geometry, Continuous Mathematics, and graduate courses in Logic and Graph Theory. More information about my teaching activities can be found on my personal website.

I work in an area of mathematical logic known as computability theory (or sometimes as recursion theory). The goal is to understand to what extent “regular” mathematics is or is not algorithmic. I am especially interested in applications to algebra and combinatorics.

I have also collaborated with a number of undergraduate students on research projects both in and out of computability theory.

I am the author of the widely adopted open source textbook Discrete Mathematics: an Open Introduction, currently in its third edition. The book is available for free as both a pdf and interactive online ebook. I have contributed to other open source textbook projects, and am an active member of the OER mathematics community.

Publications

  • Levin, O. (2025). Discrete Mathematics: an Open Introduction (4th ed.) (pp. 496). Chapman and Hall/CRC. https://doi.org/10.1201/9781003589907.
  • Kenigsberg, M., Levin, O. (2019). Prime Labelings of Infinite Graphs. Involve, a Journal of Mathematics, 12(4), 633–646. https://msp.org/involve/2019/12 4/p06.xhtml. https://doi.org/10.2140/involve.2019.12.633
  • Levin, O., McMillan, T. (2016). Computing Planarity in Computable Planar Graphs. Graphs and Combinatorics, 32(6), 2525-2539. https://link-springercom.unco.idm.oclc.org/article/10.1007%2Fs00373 016-1725-8. https://doi.org/10.1007/s00373-016-1725-8
  • Levin, O. (2016). Computable Dimension of Ordered Fields. Archive for Mathematical Logic, 55(3-4), 519-534. https://link springercom.unco.idm.oclc.org/article/10.1007%2Fs00153-016-0478-7. https://doi.org/10.1007/s00153-016-0478-7
  • Jura, M., Levin, O., Markkanen, T. (2015). A-computable graphs. Ann. Pure Appl. Logic, 167(3), 235–246. http://dx.doi.org/10.1016/j.apal.2015.11.003. https://doi.org/10.1016/j.apal.2015.11.003
  • Jura, M., Levin, O., Markkanen, T. (2015). Finding domatic partitions in infinite graphs. Electron. J. Combin., 22(3), Paper 3.39, 21. http://www.combinatorics.org/ojs/index.php/eljc/article/view/5089
  • Jura, M., Levin, O., Markkanen, T. (2014). Domatic partitions of computable graphs. Arch. Math. Logic, 53(1-2), 137–155. http://dx.doi.org/10.1007/s00153 013-0359-2. https://doi.org/10.1007/s00153-013-0359-2
  • Levin, O. (2014). Story Puzzles. College Mathematics Journal, 45(4), 296, 332-333.
  • Levin, O., Roberts, G. M. (2013). Counting knights and knaves. College Math. J., 44(4), 300–306. http://dx.doi.org/10.4169/college.math.j.44.4.300. https://doi.org/10.4169/college.math.j.44.4.300
  • Kach, A. M., Levin, O., Solomon, R. (2010). Embeddings of computable structures. Notre Dame J. Form. Log., 51(1), 55–68. http://dx.doi.org/10.1215/00294527-2010-004. https://doi.org/10.1215/00294527-2010-004
  • Flood, S., Jura, M., Levin, O., Markannen, T. The Computational Strength of Matchings in Countable Graphs.
  • Levin, O. Puzzles of Cardinality. College Mathematics Journal.
  • Grassl, R., Levin, O. (2019). Exploring Combinatorial Mathematics. http://www.openmathbooks.org/ecm/ecm.html
  • Levin, O. (2018). Discrete Mathematics: An Open Introduction, 3rd ed.. http://discrete.openmathbooks.org/dmoi3.html
  • Levin, O. (2016). Discrete Mathematics: An Open Introduction, 2nd ed. (2nd ed., pp. 327). http://discrete.openmathbooks.org
  • Levin, O. (2015). Discrete Mathematics: An Open Introduction (1st ed., pp. 296). http://discrete.openmathbooks.org
  • Levin, O., 2019 SIU Pure Mathematics Conference, “Reverse Mathematics of Maximal and Perfect Matchings,” Southern Illinois University, Carbondale, IL. (May 15, 2019).
  • Levin, O., MAA Texas Section Meeting, “Counting Knights and Knaves,” MAA Texas Section, Dallas, TX. (April 2018).
  • Levin, O., AMS Central Section Meeting, Special Session on Effective Mathematics in Discrete and Continuous Worlds, “Effective Labelings of Infinite Graphs,” AMS, Minneapolis, MN. (October 29, 2016).
  • Levin, O., Algebraic Combinatorics Seminar, “Graph Labelings and Computability Theory,” Colorado State University, Fort Collins, CO. (September 30, 2016).
  • Levin, O., University of Wyoming Colloquium, “Non computability in Graphs,,” University of Wyoming, Laramie, WY. (October 2013).
  • Levin, O., Rocky Mountain Algebraic Combinatorics Seminar, “The Computable Domatic Number of a Graph,” Colorado State University, Fort Collins, Co. (September 2013).
  • Levin, O., Frontiers of Science Institute summer program, “To Infinity and Beyond.” (June 2013).
  • Levin, O., UNC Math Teachers Circle, “Graph Theory Games.” (April 2013).
  • Levin, O. (Panelist), UNC School of Math. Sci. Pi Day, “The Great Debate about Nothing: is Zero a Natural Number?,” UNC School of Mathematical Sciences, Greeley, CO. (March 2013).
  • Levin, O., Coastal Carolina University Math Contest, “To Infinity and Beyond,” Conway, SC. (2010).
  • Levin, O., UConn Math Club, “A Surprising Theorem,” UConn Math Club. (February 2008).
  • Levin, O., UConn Math Club, “Preparing for graduate school in mathematics,” UConn Math Club. (September 2007).
  • Levin, O., Joint Mathematics Meeting, “Coloring hypergraphs is harder than coloring graphs,” AMS, Baltimore, MD. (January 17, 2019).
  • Levin, O., Joint Mathematics Meeting, “A discrete math course with early graph theory,” MAA, Baltimore, MD. (January 16, 2019).
  • Levin, O., MathFest, “Knights and Knaves and Naive Set Theory,” MAA, Denver, CO. (August 2018).
  • Levin, O., Workshop on Computability Theory and its Applications, “Measuring Complexity of Maximal Matchings of Graphs,” Fields Institute, University of Waterloo. (June 2018).
  • Levin, O., Joint Mathematics Meeting, “Prime labelings of infinite graphs and computability theory,” AMS, San Diego, CA. (January 2018).
  • Levin, O., Joint Mathematics Meeting, “Shifts in Perspectives for Counting,” MAA, San Diego, CA. (January 2018).
  • Levin, O., MAA Rocky Mountain Section Meeting, “A Paradox of Finite Cardinality,” MAA Rockey Mountain Section, Pueblo, CO. (April 2017).
  • Levin, O., Joint Mathematics Meeting, “Graph Labelings and Computability Theory,” Association for Symbolic Logic, Atlanta, GA. (January 7, 2017).
  • Boelkins, M. (Author), Levin, O. (Author & Presenter), Yoshiwara, B. (Author & Presenter), Joint Mathematics Meeting, “Interactive Textbooks in MathBookXML,” MAA, Atlanta, GA. (January 7, 2017).
  • Levin, O., Joint Mathematics Meeting, “Tricks to Make Counting Harder for Students,” MAA, Atlanta, GA. (January 7, 2017).
  • Levin, O., Joint Mathematics Meeting, “”Finishing” an Open Textbook,” MAA, Seattle, WA. (January 2016).
  • Levin, O., McMillan, T. (Presenter), Joint Mathematics Meeting, “Embeddings of Computable Planar Graphs,” AMS, Seattle, WA. (January 2016).
  • Jura, M., Levin, O. (Presenter), Markkanen, T., Joint Mathematics Meeting, “Graphs between Computable and Highly Computable,” AMS, Seattle, WA. (January 2016).
  • Levin, O., Joint Mathematics Meeting, “Knights and Knaves in the Classroom,” MAA, Seattle, WA. (January 2016).
  • Levin, O. (Presenter), Seehausen, A. T., MAA Rocky Mountain Section Meeting, “Using Proofs to Introduce Logic,” MAA, Colorado Springs, CO. (April 2015).
  • Levin, O., Myrant, C. (Presenter), Joint Mathematics Meeting, “Coloring Around Faces to Count Daisies,” MAA, San Antonio, TX. (January 2015).
  • Jura, M., Levin, O. (Presenter), Markkanen, T., Joint Mathematics Meeting, “Controlling Domination in Infinite Graphs,” AMS, San Antonio, TX. (January 2015).
  • Levin, O., ASL North American Annual Meeting, “Finding Small Domatic Partitions in Graphs with Large Domatic Number,” Association of Symbolic Logic, Boulder, CO. (May 2014).
  • Jura, M. (Presenter), Levin, O., Markkanen, T., Joint Mathematics Meeting, “Domatic Partitions of A-Computable Graphs and c.e. Permitting,” AMS, Baltimore, MD. (January 2014).
  • Jura, M., Levin, O., Markkanen, T. (Presenter), Joint Mathematics Meeting, “Highly Computable Graphs and their Domatic Numbers,” ASL, Baltimore, MD. (January 2014).
  • Levin, O., Joint Mathematics Meeting, “The Complexity of Transcendence Bases in Computable Ordered Fields,” AMS, Baltimore, MD. (January 2014).
  • Jura, M., Levin, O., Markkanen, T. (Presenter), Joint Mathematics Meeting, “Domatic Partitions of Computable Graphs,” ASL, San Diego, CA. (January 2013).
  • Levin, O., MAA Rocky Mountain Section Meeting, “How (Not) to Compute Domatic Partitions of Graphs,” MAA, Denver, CO. (April 2012).
  • Levin, O., Joint Mathematics Meeting, “Counting Liars and Truth-tellers: Binomial Identities from Logic Puzzles,” MAA, Boston, MA. (January 2012).
  • Levin, O., Joint Mathematics Meeting, “Computable Dimension of Ordered Fields,” MAA. (January 2009).
  • Levin, O., New York Graduate Student Logic Conference, “Goodstein’s Theorem and the Limits of PA,” New York, NY. (2007).
  • Levin, O., Section Meeting, “Twin Relative Primes,” MAA Rocky Mountain, Colorado Springs, CO. (April 2003).
  • Levin, O., UNC SMS Seminar, “Maximal Matchings and Reverse Mathematics,” UNC School of Mathematical Sciences, Greeley, CO. (April 2016).
  • Levin, O., UNC SMS Seminar, “Knights and Knaves in the Classroom,” UNC School of Mathematical Sciences, Greeley, CO. (November 2015).
  • Levin, O., UNC SMS Seminar, “Measuring Set Complexity for Graph Coloring,” UNC School of Mathematical Sciences, Greeley, CO. (April 2015).
  • Levin, O. (Author & Presenter), UNC SMS Seminar, “Effective Graph Theory,” UNC School of Mathematical Sciences, Greeley, CO. (October 2013).
  • Levin, O., UNC Math. Ed. Seminar, “Tricks to Make Counting Harder for Students.” (April 2013).
  • Levin, O., UNC Math. Ed. Seminar, “How to Not Teach Proofs.” (November 2012).
  • Levin, O., UConn Math Club, “Graph Theory and the Four Color Theorem,” UConn Math Club, Storrs, CT. (September 2008).
  • Levin, O., SIGMA Seminar, “Goodstein’s Theorem,” University of Connecticut. (March 2007).
  • Levin, O., SIGMA Seminar, “Not a Proof of the Four Color Theorem,” University of Connecticut. (September 2006).
  • Levin, O., SIGMA Seminar, “Games Mathematicians Play,” University of Connecticut. (April 2006).
  • Levin, O., SIGMA Seminar, “Ramsey Theory,” University of Connecticut. (April 2005).

Grants and Sponsored Research

  • Levin, O., Safranski, C., Siefken, J., “PreTeXt for Small Documents workshop,” Sponsored by American Institute of Mathematics, Federal.
  • Levin, O. (Principal), Furst, V. (Principal), McKenna, P. A. (Local Organizer), “Collaborative OER Mathematics Textbooks (COMaT),” Sponsored by CDHE, State, $51,916.00. (April 2024 – June 2025).
  • Levin, O., “Workshop on Computability Theory and its Applications,” Sponsored by Provost Award for Travel, University of Northern Colorado, $1,115.00. (April 25, 2018 – June 30, 2018).
  • Levin, O. (Supporting), “Northern Colorado Mathematics Collaboration to Support Teacher Quality Project,” Sponsored by Colorado Department of Higher Education, State, $161,228.00 (2013 – 2014).
  • Levin, O. (Principal), “Internal Faculty Reassignment Award for Research Scholarship and Creative Works,” Sponsored by University of Northern Colorado, University of Northern Colorado, $4,800.00. (2013 – 2014).