PLANAR LATTICE GRAPHS WITH.dvi

Introduction. The property of the Petersen graph of being hypohamiltonian is notorious, and known for a long time. The existence of hypotraceable graphs, on the contrary, was discovered much later. Before that, in 1966, Gallai [1] had asked the easier question whether (connected) graphs do exist such that each vertex is missed by some longest path. Such graphs have been discovered by Walther in 1969 [6] before J. Horton found the first hypotraceable graph. In this context appeared Zamfirescu’s questions about the existence of (small, if possible minimal) k-connected graphs with the property that for any j vertices there is a longest path avoiding all of them [9]. The questions were also asked for cycles instead of paths, and, on the other hand, specialized to planar graphs. Since several good examples answering Zamfirescu’s questions were published in the following years by Grünbaum [2], Schmitz [5], Zamfirescu [10], [11], Zamfirescu and Zamfirescu [8] and others, an opposite question was asked: Are there large families of graphs without any such examples? B. Menke discovered one such family, the grid graphs. These are (finite) subgraphs of the infinte square lattice graph L in the plane consisting of all vertices and all edges lying on or within a cycle in L. Menke proved that every grid graph has at least 4 vertices lying on every longest cycle ([3], see also [4]). Motivated by this result, we investigate here other graphs embedded in L concerning the properties mentioned above. Moreover, we shall extend our investigation to graphs embedded in the (infinite) hexagonal lattice H. We find out that graphs with the mentioned properties do exist (for j = 1), although their orders are considerably larger.