Multiplicative property of virtual edges:

Let \(G_1\) be a graph with chromatic number \(c_0\) which virtualizes an edge length \(d_1\). Then replacing all edges of \(G_0\) with virtual edges of length \(d_1\) produces a virtual edge of chromatic color \(c_0\) with length \(d_0*d_1\).

If a graph, \(G_2\), with chromatic number \(c_2\geq 1+c_0\) is created using virtual edges with chromatic number \(c_0\), then a graph, \(G_3\), of chromatic number \(c_3\) (s.t. \(c_2\geq c_3\geq 1+c_0\)) can be created by replacing the virtual edges with the graphs which virtualize the virtual edges.

More trivially, if a graph, \(G_4\), with chromatic number \(c_4\leq c_0\) is created using virtual edges with chromatic number \(c_0\), then replacing the virtual edges with the graphs which virtualize the virtual edges produces a graph with chromatic number \(c_0\).

If virtual edges with chromatic number 4 are found, then they may be useful to make a graph of chromatic number 5 which uses fewer vertexes than the current smallest graph. In an extreme example, 5 vertices would be enough if each of the distances between pairs of vertices corresponds to a virtual edge with chromatic number 4.

]]>May I ask you a little stupid question that you have any works related to Hadwiger-Nelson problem? Thank for your response. ]]>