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This machine essentially uses cycles of length 2, whereas the first example had only 1-cycles.
Let X r be the number of cycles of length r in G.
Color intensity represents the number of cycles of length ℓ linking n communities.
A bipartite graph is chordal bipartite if each of its cycles of length, at least 6, has a chord.
k = 3 is possible for all such graphs (k = 2 is not sufficient as seen for instance in complete graphs and cycles of length not divisible by 4).
As an example shown in Figure 2, there are three types of cycles of length 4, and only the third one is a simple cycle.
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Let (Ginmathbb{U}(n)) with unique cycle of length (m ge4), where (nge2m).
A chordal graph is a graph that contains no induced cycle of length four or greater.
Let (Ginmathbb{U}(n)) with unique cycle of length (m ge3), where (n ge2m).
If C is a cycle of length at most 4, then we are done.
Let (Ginmathbb{U}(n)) with unique cycle of length m, where (nge2mge8).
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