Tur\'an numbers of vertex-disjoint cliques in $r$-partite graphs

TL;DR

This paper determines the maximum number of edges in subgraphs of complete r-partite graphs that avoid containing k disjoint r-cliques, extending previous results to a broader class of graphs.

Contribution

It generalizes the Turán number for vertex-disjoint cliques in r-partite graphs, covering all r ≥ 3 and k ≥ 1, which was previously known only for bipartite cases.

Findings

Derived formulas for Turán numbers in r-partite graphs

Extended previous bipartite results to r-partite graphs

Provided comprehensive solutions for all r ≥ 3 and k ≥ 1

Abstract

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this paper we will determine the Tur\'{a}n numbers $ex(K_{a_1,\ldots,a_r},kK_r)$ for all $r\geq 3$ and $k\geq 1$.