Publication details
On possible counterexamples to Negami's planar cover conjecture
| Authors | |
|---|---|
| Year of publication | 2004 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Graph Theory |
| MU Faculty or unit | |
| Citation | |
| Web | http://www3.interscience.wiley.com/cgi-bin/fulltext/108061224/ABSTRACT |
| Field | General mathematics |
| Keywords | graph; planar cover; projective plane; minor |
| Description | A simple graph $\H$ is a cover of a graph $\G$ if there exists a mapping $\varphi$ from $\H$ onto $\G$ such that $\varphi$ maps the neighbors of every vertex $v$ in $\H$ bijectively to the neighbors of $\varphi(v)$ in $\G$. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, and the first author that the conjecture holds as long as the graph $\K_{1,2,2,2}$ has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most $16$ possible counterexamples to Negami's conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list. |