Bipartite divisor graphs for integer subsets

TL;DR

This paper introduces the bipartite divisor graph for integer subsets, exploring its properties and relationships with other divisor graphs, and characterizes the graphs that can arise from such sets.

Contribution

It defines the bipartite divisor graph for integer sets and establishes its connections with common divisor and prime vertex graphs, including characterizations and conditions for specific subgraphs.

Findings

Links between graph parameters such as number and diameter of components.

Characterization of bipartite graphs that can be realized as B(X).

Necessary and sufficient conditions for the presence of complete subgraphs.

Abstract

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph $\Ga(X)$ and the prime vertex graph $\Delta(X)$, for a set $X$ of positive integers, we define the bipartite divisor graph $B(X)$, and show that many of these connections flow naturally from properties of $B(X)$. In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as $B(X)$ for some $X$. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of $B(X)$, for one $\Gamma(X)$ or $\Delta(X)$ to contain a complete subgraph of size 3 or 4.