On the minimal degree condition of graphs implying some properties of subgraphs

TL;DR

This paper investigates minimum degree conditions in graphs that guarantee certain subgraph properties, generalizing Erdős's problem and establishing bounds for parameters related to r-partite and r-clique-free subgraphs.

Contribution

It generalizes Erdős's problem to broader cases, providing bounds on minimum degree thresholds for properties involving r-partite and r-clique-free subgraphs.

Findings

Derived bounds for _r for r  4

Established _4  0.9415

Generalized conditions for subgraph properties in graphs

Abstract

Erd\H{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let $\delta_r$ be the least number so that any graph $G$ on $n$ vertices with minimum degree $\delta_rn$ has the property $P_{r-1}(G)=K_rf(G),$ where $P_{r-1}(G)$ is the largest number of edges in an $(r-1)$-partite subgraph and $K_rf(G)$ is the largest number of edges in a $K_r$-free subgraph. We show that $\frac{3r-4}{3r-1}<\delta_r\le\frac{4(3r-7)(r-1)+1}{4(r-2)(3r-4)}$ when $r\ge4.$ In particular, $\delta_4\le 0.9415.$