Extremal numbers of disjoint triangles in $r$-partite graphs

TL;DR

This paper extends classical extremal graph theory results by determining the maximum number of edges in r-partite graphs that avoid k disjoint triangles, under specific size constraints.

Contribution

It provides the first known extremal number results for disjoint triangles in r-partite graphs with r ≥ 4 and particular size conditions.

Findings

Determined ex$(K_{n_1,n_2, dots,n_r},kK_3)$ for r ≥ 4.

Extended classical extremal results to new multipartite graph configurations.

Established bounds for the extremal number under given size constraints.

Abstract

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a classical extremal problem in graph theory. In 1962, Erd\H{o}s determined {ex}$(K_n,kK_3)$, which generalized Mantel's Theorem. On the other hand, in 1974, {Bollob\'{a}s}, Erd\H{o}s, and Straus determined {ex}$(K_{n_1,n_2,\dots,n_r},K_t)$, which extended Tur\'{a}n's Theorem to complete multipartite graphs. { In this paper,} we determine {ex}$(K_{n_1,n_2,\dots,n_r},kK_3)$ for $r\ge 4$ and $10k-4\le n_1+4k\le n_2\le n_3\le \cdots \le n_r$.