| Research Abstract |
Negami, the head investigator of this project, proposed in 1986 the conjecture that if a connected graph has a finite planar covering, then it can be embeddable in the projective plane and many topological graph theorists over the world support his conjecture, calling "Planar Cover Conjecture". Their studies up to the present conclude that if K_<1,2,2,2> has no finite planar coverings, the conjecture is true. However, it is so difficult to prove that such a graph has no finite planar covering since it has infinitely many coverings among which a finite planar one might hide. So we tried to establish a theoretical upper bound for the number of coverings which we should search to decide whether or not there exists a finite planar covering and a method to generate such a finite search space efficiently. Such a research strategy is called "finitizing Planar Cover Conjecture".
For generating the search space, we established an O(n) time algorithm which decides whether or not there exists an n-fold planar covering of K_<1,2,2,2>, assuming the non-existence of its (n-1)-fold planar covering. However, its memory space becomes so big that a program simply implemented on PC would not run consistently. This suggests a future study on "designing a sustainable system" in an aspect of information sciences.
To establish a theory to bound the size of search spaces, we have discussed it with a conjecture that sufficiently large planar coverings will be composite and found the fact that it happens very often, linking our theory on composite coverings and that on primitive permutation groups in group theory, which shows the efficiency of research on finite groups and zeta functions of graphs, which can be regarded as a kind of a generating function of cycles in graphs. Furthermore, we could establish many theorems on other topics in topological graph theory, being motivated by this project.
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