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Note on vertex disjoint rainbow triangles in edge-colored graphs

Jürgen Kritschgau, tahda queer, Cyrus Young, Wohua Zhou

Abstract

Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow triangles. Rainbow problems have been studied extensively in the context of anti-Ramsey theory, and more recently, in the context of Turán problems. B. Li. et al. \textit{European J. Combin. 36 (2014)} found that a graph must contain a rainbow triangle if $e(G)+c(G) \geq \binom{n}{2}+ n$. L. Li. and X. Li. \textit{Discrete Applied Mathematics 318 (2022)} conjectured a lower bound on $e(G)+c(G)$ such that $G$ must contain a proper $mK_3$. In this paper, we provide a construction that disproves the conjecture. We also introduce a result that guarantees the existence of $m$ vertex disjoint rainbow $K_k$ subgraphs in general host graphs, and a sharp result on the existence of proper $mK_3$ in complete graphs.

Note on vertex disjoint rainbow triangles in edge-colored graphs

Abstract

Given an edge-colored graph , we denote the number of colors as , and the number of edges as . An edge-colored graph is rainbow if no two edges share the same color. A proper is a vertex disjoint union of rainbow triangles. Rainbow problems have been studied extensively in the context of anti-Ramsey theory, and more recently, in the context of Turán problems. B. Li. et al. \textit{European J. Combin. 36 (2014)} found that a graph must contain a rainbow triangle if . L. Li. and X. Li. \textit{Discrete Applied Mathematics 318 (2022)} conjectured a lower bound on such that must contain a proper . In this paper, we provide a construction that disproves the conjecture. We also introduce a result that guarantees the existence of vertex disjoint rainbow subgraphs in general host graphs, and a sharp result on the existence of proper in complete graphs.
Paper Structure (4 sections, 7 theorems, 10 equations)

This paper contains 4 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Lower Bound Constructions
  3. Proof of Proposition \ref{['thm:main1']}
  4. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

(Theorem 1 in li2014rainbow). Let $G$ be an edge-colored graph on $n$ vertices. If $e(G) + c(G) \geq \binom{n+1}{2}$, then $G$ contains a rainbow triangle.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Proposition 1.5
  • Theorem 1.6
  • proof : Proof of Proposition \ref{['thm:main1']}
  • Claim 3.1
  • proof
  • Lemma 4.2
  • ...and 10 more