A Note on d-Biclique Covers

TL;DR

This paper investigates the d-biclique covering number of graphs, providing upper bounds for lexicographic products, bounds for various graph constructions, and exact values for specific graphs.

Contribution

It introduces new bounds for the d-biclique covering number in lexicographic graph products and other constructions, along with exact calculations for particular cases.

Findings

Established an upper bound for the d-biclique covering number of lexicographic product graphs.

Derived bounds for the d-biclique covering number in various graph constructions.

Determined the exact d-biclique covering number for specific classes of graphs.

Abstract

A d-biclique cover of a graph G is a collection of bicliques of G such that each edge of G is in at least d of the bicliques. The number of bicliques in a minimum d-biclique cover of G is called the d-biclique covering number of G and is denoted by ${bc}_d(G)$. In this paper, we present an upper bound for the d- biclique covering number of the lexicographic product of graphs. Also, we introduce some bounds of this parameter for some graph constructions and obtain the exact value of the d-biclique covering number of some graphs.