Clique-partitioned graphs

TL;DR

This paper characterizes the structure of graphs that can be uniquely partitioned into a fixed number of cliques, and identifies those with the maximum number of edges within these constraints.

Contribution

It provides a complete structural characterization of weakly and strongly clique-partitioned graphs with maximum edges.

Findings

Identifies the maximum number of edges in such graphs.

Provides a unique decomposition into cliques for these graphs.

Characterizes the structure of extremal clique-partitioned graphs.

Abstract

A graph $G$ of order $nv$ where $n\geq 2$ and $v\geq 2$ is said to be weakly $(n,v)$-clique-partitioned if its vertex set can be decomposed in a unique way into $n$ vertex-disjoint $v$-cliques. It is strongly $(n,v)$-clique-partitioned if in addition, the only $v$-cliques of $G$ are the $n$ cliques in the decomposition. We determine the structure of such graphs which have the largest possible number of edges.