# Maximizing the number of vertices of an N-clique cover of the edges of a graph on N vertices

## TL;DR

This paper investigates the maximum total size of cliques covering a graph with a fixed number of cliques equal to the number of vertices, providing upper bounds for such coverings.

## Contribution

It introduces a new perspective by fixing the number of cliques to the number of vertices and establishing upper bounds for the sum of their sizes.

## Key findings

- Derived upper bounds for clique coverings with equal number of vertices and cliques.
- Extended understanding of edge-covering strategies beyond minimal clique covers.
- Provided conditions under which such coverings are possible.

## Abstract

All the work made so far on edge-covering a graph by cliques focus on finding the minimum number of cliques that cover the graph. On this paper, we fix the number of cliques that cover a graph by the same number of vertices that the graph has, and give an upper bound for the sum of the number of vertices of these cliques in the cases where this covering is possible.