Hamiltonian Circuit Problem

The most elementary graph is tree. Tree is a connected graph with no cycles. Tree is able to represent hierarchy structure and admittedly almost things and systems have this tree structures. Then what is structure or where does it come from?  It can be considered that every thing on the ground is additive, while additivity is the principle of the world where such a calculation as 1+1=2 holds. It is the finite set theoretical world. Due to that every thing has additive characteristic, it becomes able to partition them, then classify them. Tree is a presentation form of the world where such additivity stands.

Aristotle is the first that aimed at such character of the world. Aristotle (384-322B.C.) thought that the world is completely describable by classifying things and recomposing it. Descartes (1596-1650) showed in Discours de la Methode that such principle of additivity is adaptable for not only things but also methods. He thought that by partitioning a process, and refining the partition repeatedly, a process finally turns out to be recomposed in a tree structure having leaves of very simple unit process able to be performed at once. Time needed  from Aristotle to Descartes was about 2000 years. From Descartes to Henry Ford (1863-1947) only needs 250 years. Anyway these two concepts/methods are the foundation which supported the whole weight of Western Natural Science. Then look back and find why the East had dropped behind in natural science. For instance, Indian Philosophy based on Causality Rule has an appetite for natural science. Apart from propositional logic invented by Greek, Indians devised their logical philosophy independently. Why the East failed to give birth to Science or Mathematics? It was almost sure that she had conceived it anyhow.

Now we are in the 21th Century and apparently our civilization has reached at the next stage where it seems to demand a graph theoretical understanding of the world. Graph Theory is a study regarding relations, and you may recognize a clear trace where  Eastern Philosophy had intended to deal with "relation" from the beginning. An adjective "intractable" appears often in the complexity theory books with regards to complexity of calculation, then we can say that they carried intractable problems on their back from its starting point because the cause of any intractability is clearly due to an embedded circulation in the problem. If it is so, it may be the time to read Eastern Philosophy again in a bit fresh eyes. Briefly speaking, a graph is to be seen as a compound of trees and cycles, and now we understand that the tree was planted on the West ground and the cycle was circulated in the Eastern river. I would like to say that hierarchy is the masculine principle and circulation is the feminine principle. The reconciliation of  both principles may be the meaning of this century. [rev. 2005-01-22]