Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nearly tight bound for rainbow clique subdivisions in properly edge-colored graphs and applications

Peiru Kuang, Yan Wang

TL;DR

We study rainbow analogues of subdivision problems in edge-colored graphs, focusing on rainbow subdivisions of cliques. The authors develop a robust sublinear expander framework and a two-stage thinning–sprinkling method to force rainbow clique subdivisions with controlled path lengths, showing that a graph on $n$ vertices with average degree at least $C t^2 \log n (\log \log n)^6$ contains a rainbow $TK_t$ where each edge of $K_t$ is replaced by a path of length $O(\log n \cdot \log \log n)$. For fixed $t$, this bound is essentially tight, resolving a key question on rainbow clique subdivisions and yielding a rainbow-cycle consequence. The results have broad applications in additive combinatorics, number theory, and coding theory, including bounds on dissociated sets, $B_h[g]$-sequences in groups, and dimensions of locally correctable codes.

Abstract

An edge-colored graph is said to be rainbow if all its edges have distinct colors. In this paper, we study the rainbow analogue of a fundamental result of Mader [\emph{Math. Ann.} \textbf{174} (1967), 265--268] on the existence of subdivisions in graphs with large average degree. This is part of the study of rainbow analogues of classical Turán problems, a framework systematically introduced by Keevash, Mubayi, Sudakov and Verstraëte [\emph{Combin. Probab. Comput.} \textbf{16} (2007), 109--126]. We prove that every properly edge-colored graph on $n$ vertices with average degree at least $t^2(\log n)^{1+o(1)}$ contains a rainbow subdivision of $K_t$. When $t$ is a constant, this bound is tight up to the $o(1)$ term. So it essentially resolves a question raised by Jiang, Methuku and Yepremyan [\emph{European J. Combin.} \textbf{110} (2023), 103675] on rainbow clique subdivisions, and also implies a result of Alon, Bucić, Sauermann, Zakharov and Zamir [\emph{Proc. Lond. Math. Soc.} \textbf{130} (2025), e70044] on rainbow cycles. In addition, we present several applications of our result to problems in additive combinatorics, number theory and coding theory.

Nearly tight bound for rainbow clique subdivisions in properly edge-colored graphs and applications

TL;DR

We study rainbow analogues of subdivision problems in edge-colored graphs, focusing on rainbow subdivisions of cliques. The authors develop a robust sublinear expander framework and a two-stage thinning–sprinkling method to force rainbow clique subdivisions with controlled path lengths, showing that a graph on vertices with average degree at least contains a rainbow where each edge of is replaced by a path of length . For fixed , this bound is essentially tight, resolving a key question on rainbow clique subdivisions and yielding a rainbow-cycle consequence. The results have broad applications in additive combinatorics, number theory, and coding theory, including bounds on dissociated sets, -sequences in groups, and dimensions of locally correctable codes.

Abstract

An edge-colored graph is said to be rainbow if all its edges have distinct colors. In this paper, we study the rainbow analogue of a fundamental result of Mader [\emph{Math. Ann.} \textbf{174} (1967), 265--268] on the existence of subdivisions in graphs with large average degree. This is part of the study of rainbow analogues of classical Turán problems, a framework systematically introduced by Keevash, Mubayi, Sudakov and Verstraëte [\emph{Combin. Probab. Comput.} \textbf{16} (2007), 109--126]. We prove that every properly edge-colored graph on vertices with average degree at least contains a rainbow subdivision of . When is a constant, this bound is tight up to the term. So it essentially resolves a question raised by Jiang, Methuku and Yepremyan [\emph{European J. Combin.} \textbf{110} (2023), 103675] on rainbow clique subdivisions, and also implies a result of Alon, Bucić, Sauermann, Zakharov and Zamir [\emph{Proc. Lond. Math. Soc.} \textbf{130} (2025), e70044] on rainbow cycles. In addition, we present several applications of our result to problems in additive combinatorics, number theory and coding theory.
Paper Structure (15 sections, 24 theorems, 74 equations)

This paper contains 15 sections, 24 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Rainbow cycles
  3. Rainbow subdivisions
  4. Applications
  5. Additive combinatorics
  6. Number theory
  7. Coding theory
  8. Proof overview
  9. Robust sublinear expanders
  10. Colorful expansion of random samples
  11. Expansion of small sets: Sprinkling
  12. Expansion of large sets: Thinning
  13. Applications
  14. Additive combinatorics
  15. Number theory

Key Result

Theorem 1.4

Let $n$ and $t$ be integers. Let $G$ be a properly edge-colored graph on $n$ vertices with $d(G)\geq C t^2\cdot \log n \cdot (\log \log n)^{6}$, where $C$ is a sufficiently large constant. Then $G$ contains a rainbow $TK_t$, where each edge of $K_t$ is replaced by a path of length at most $O(\log n\

Theorems & Definitions (52)

  • Definition 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 1.10
  • Theorem 1.11
  • Definition 1.12
  • ...and 42 more