HIGHLY NON-CONCURRENT LONGEST.dvi

There exist planar graphs in which any two vertices are missed by some longest cycle. Although this requirement is very strong, we prove here that such graphs can also be found as subgraphs of the square and hexagonal lattices. Considering (finite) such lattices on the torus and on the Möbius strip enables us to reduce the order of our examples. Introduction Do we have small (if possible minimal) k-connected graphs with the property that for any j vertices there is a longest path (cycle) avoiding all of them? This question was a strengthening of Gallai’s question [1] and it was raised by Zamfirescu [8]. In particular, the question was (also) asked for planar graphs. After getting several answers for his questions (see [7], [3], [9]), in 2001 he asked to investigate lattice graphs from this point of view [10]. We shall say that a graph is a C k -graph if it is k-connected and any set of j vertices is missed by some longest cycle. The existence of C 2 graphs in various lattices has been verified in [2], [4] and [5]. So far, no C 2 2 -graphs have been discovered in any lattice. It is perhaps interesting to mention that no C 2 -graph at all is known, so far. In this paper, we first prove the existence of C 2 -graphs in the infinite square lattice L and hexagonal lattice H, obviously with a considerably larger number of vertices than the smallest known C 2 -graph in the plane, whose order is 135, found by Zamfirescu [9]. Second, we consider (finite) square and hexagonal lattices on the torus and on the Möbius strip, which are defined according to [4], and construct C 2 2 -subgraphs of these lattices. 2 Ayesha Shabbir, Tudor Zamfirescu Of great help will be the following lemma, due to which proofs of our main results reduce to finding appropriate embeddings in the various lattices considered here.