Anti-Ramsey numbers for vertex-disjoint triangles

TL;DR

This paper determines the maximum number of colors in an edge-coloring of a complete graph that avoids rainbow disjoint triangles, providing exact values for certain ranges and bounds for others.

Contribution

It establishes the anti-Ramsey number for vertex-disjoint triangles in complete graphs for specific ranges of n and k, filling gaps in existing knowledge.

Findings

Exact anti-Ramsey numbers for n=3k and n≥2k²−k+2.

Lower and upper bounds for intermediate n values.

New results extend understanding of rainbow triangle configurations.

Abstract

An edge-colored graph is called rainbow if all the colors on its edges are distinct. Given a positive integer n and a graph G, the anti-Ramsey number ar(n,G) is the maximum number of colors in an edge-coloring of K_{n} with no rainbow copy of G. Denote by kC_{3} the union of k vertex-disjoint copies of C_{3}. In this paper, we determine the anti-Ramsey number ar(n,kC_{3}) for n=3k and n\geq2k^{2}-k+2, respectively. When 3k\leq n\leq 2k^{2}-k+2, we give lower and upper bounds for ar(n, kC_{3}).