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Bounds for rainbow-uncommon graphs

Blake Bates, Zhanar Berikkyzy, Nick Chiem, Gabriel Elvin, Risa Fines, Maja Lie, Hanna Mikulás, Isaac Reiter and Kevin Zhou
Vol. 19 (2026), No. 2, 249–257
DOI: 10.2140/involve.2026.19.249
Abstract

We say a graph H is r-rainbow-uncommon if the maximum number of rainbow copies of H under an r-coloring of E(Kn) is asymptotically (as n →∞) greater than what is expected from uniformly random r-colorings. Via explicit constructions, we show that K3 is r-rainbow-uncommon for r ≥ 3. We also construct colorings to show that for t ≥ 4, Kt is r-rainbow-uncommon for sufficiently large r.

Keywords
anti-Ramsey multiplicity, rainbow coloring, common/uncommon
Mathematical Subject Classification
Primary: 05A16, 05C15, 05D10
Milestones
Received: 13 February 2024
Revised: 6 August 2024
Accepted: 14 August 2024
Published: 8 March 2026

Communicated by Kenneth S. Berenhaut
Authors
Blake Bates
Department of Mathematics
University of Arizona
Tucson, AZ
United States
Zhanar Berikkyzy
Mathematics Department
Fairfield University
Fairfield, CT
United States
Nick Chiem
Department of Mathematics
University of California, Riverside
Riverside, CA
United States
Gabriel Elvin
Department of Mathematics
California State University, San Bernardino
San Bernardino, CA
United States
Risa Fines
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Maja Lie
Department of Mathematics
Imperial College London
London
United Kingdom
Hanna Mikulás
Minerva University
San Francisco, CA
United States
Isaac Reiter
Department of Mathematics
Lehigh University
Bethlehem, PA
United States
Kevin Zhou
Department of Mathematics
University of Illinois Urbana-Champaign
Urbana, IL
United States
© 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).