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The anti-Ramsey numbers of cliques in complete multi-partite graphs

Yuyu An, Ervin Gyori, Binlong Li

Abstract

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Let $G$ and $H$ be two graphs. The anti-Ramsey number $\ar(G, H)$ is the maximum number of colors of an edge-coloring of $G$ that does not contain a rainbow copy of $H$. In this paper, we study the anti-Ramsey numbers of $K_k$ in complete multi-partite graphs. We determine the values of the anti-Ramsey numbers of $K_k$ in complete $k$-partite graphs and in balanced complete $r$-partite graphs for $r\geq k$.

The anti-Ramsey numbers of cliques in complete multi-partite graphs

Abstract

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Let and be two graphs. The anti-Ramsey number is the maximum number of colors of an edge-coloring of that does not contain a rainbow copy of . In this paper, we study the anti-Ramsey numbers of in complete multi-partite graphs. We determine the values of the anti-Ramsey numbers of in complete -partite graphs and in balanced complete -partite graphs for .
Paper Structure (9 sections, 12 theorems, 8 equations)

This paper contains 9 sections, 12 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Symmetrization of graphs and colorings
  4. Blow-up of graphs and colorings
  5. Proofs of Theorems \ref{['ThkPartite']} and \ref{['ThBalanced']}
  6. The extremal colorings
  7. For $K_k$ in complete $k$-partite graphs
  8. For $K_k$ in balanced $r$-partite graphs
  9. Acknowledgements

Key Result

Theorem 1.1

For all $n\geq 3$, ${\rm ar}(K_n,K_3)=n-1$.

Theorems & Definitions (28)

  • Theorem 1.1: Erdős et al. ErSiSo
  • Theorem 1.2: Schiermeyer Sc
  • Theorem 1.3: Fang et al. FaGyLiXi
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 18 more