A note on the clique number of complete $k$-partite graphs

TL;DR

This paper proves that among all degree-equivalent simple graphs, the complete k-partite graph uniquely has a clique number of k, providing a sharper lower bound on clique number for a broad class of graphs.

Contribution

It establishes the uniqueness of complete k-partite graphs in having clique number k among degree-equivalent graphs, improving existing bounds.

Findings

Complete k-partite graph is unique with clique number k among degree-equivalent graphs

Provides a sharper lower bound on clique number for a large family of graphs

Enhances understanding of the relationship between degree sequences and clique numbers

Abstract

In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.