2025/11/06 by Xiaonan Liu, Liu, Xiaonan
Computer Science · Mathematics · #05C10 #05C15 #05C35 #05C38 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2511.04066
openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28
The rainbow Turán number of a fixed graph H, denoted by ex^*(n,H), is the maximum number of edges in an n-vertex graph such that it admits a proper edge coloring with no rainbow H. We study this problem in planar setting. The rainbow planar Turán number of a graph H, denoted by exP^*(n,H), is the maximum number of edges in an n-vertex planar graph such that it has a proper edge coloring with no rainbow H. We consider the rainbow planar Turán number of cycles. Since C3 is complete, exP^*(n, C3) is exactly its planar Turán number, which is 2n-4 for n≥ 3. We show that exP^*(n, C4)=3n-6 for n=k2-3k+2 where k≥ 5, and exP^*(n,Ck)=3n-6 for all k≥ 5 and n≥ 3.