A simple proof of the Gan-Loh-Sudakov conjecture

TL;DR

This paper presents a new, unified proof of a conjecture relating to the maximum number of cliques of a certain size in simple graphs with bounded maximum degree, providing a clearer understanding of graph clique structures.

Contribution

The paper offers a simplified, unified proof of the Gan-Loh-Sudakov conjecture on clique counts in graphs with bounded degree.

Findings

Proves the maximum number of t-cliques in graphs with degree at most Δ

Provides a unified proof approach for the conjecture

Clarifies the relationship between graph parameters and clique counts

Abstract

We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $\Delta$ has no more than $a\binom{\Delta+1}{t}+\binom{b}{t}$ cliques of size $t \ (t \ge 3)$, where $n = a(\Delta+1)+b \ (0 \le b \le \Delta)$.