Another Two Graphs With No Planar Covers

. A graph H is a cover of a graph G if there exists a mapping ' from V (H) onto V (G) such that for every vertex v of G, ' maps the neighbors of v in H bijectively to the neighbors of '(v) in G. Negami conjectured in 1986 that a connected graph has a nite planar cover if and only if it embeds in the projective plane. It follows from the results of Archdeacon, Fellows, Negami, and the author that the conjecture holds as long as K 1;2;2;2 has no nite planar cover. However, this is still an open question, and K 1;2;2;2 is not the only minor-minimal graph in doubt. Let C4 (E 2) denote the graph obtained from K 1;2;2;2 by replacing two vertex-disjoint triangles (four edgedisjoint triangles) not incident with the vertex of degree 6 with cubic vertices. We prove that the graphs C4 and E 2 have no planar covers. This fact is used in [P. Hlineny, R. Thomas, On Possible Counterexamples to Negami's Planar Cover Conjecture, manuscript 1999] to show that that there are, up to obvious construct..