Bipartite divisor graphs for integer subsets

Details Bipartite divisor graphs for integer subsets Bipartite divisor graphs for integer subsets

2009-10-28

arxiv.org/abs/0910.5396v1

Mathematics

Combinatorics

Scientific paper

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.

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