Fans and bundles in the graph of pairwise sums and products (Q1422145)

In the present paper a graph \(G\) is defined where the vertex set is the positive integers, and the set of the edges is defined as follows: for two different integers \(n\) and \(m\), \((n,m)\) is an edge, if there exist integers \(x\) and \(y\) such that \(n=x+y\) and \(m=x\cdot y\). (In the paper it is not explicitly mentioned that this graph is clearly a directed graph). Since for every positive \(t\), \(t<n\) the ordered sequence \(n,n-1,\dots ,n-t\) is a directed path in the graph the author introduces the notion of \(t\)-fan: there exists a vertex \(m\), such that for every \(i\), \(0\leq i\leq n-t\), \((i,m)\) is an edge. As the first result of the paper, the author characterizes \(t\)-fans of the graph. A subgraph of \(G\) is said to be a \(t\)-bundle if for some \(n\) there exist integers \(n_1,n_2,\dots ,n_t\) such that both \((n,n_i)\), \(i=1,2,\dots, t\) and \((n-1,n_i)\), \(i=1,2,\dots, t\) are edges. It is proved that for every integer \(t\) \(G\) contains a \(t\)-bundle. Furthermore it is proved that the chromatic number of the graph \(G\) is at least \(4\). Some related questions are also discussed.