Highly nonconcurrent longest paths and cycles in lattices

Abstract

We investigate here the connected graphs with the property that any pair of vertices are missed by some longest paths (or cycles), embeddable in n -dimensional lattices L n where L denotes the set of integers.

Highly non-concurrent longest paths in lattices

“…Figure 5 shows such an embedding of G in T . For a P 2 1 -graph in L , see Figure 6; it is of order 617 and is obtained under the conditions of Lemma…”

“…We consider an (m + 1) × (n + 1) parallelogram (with (m + 1)(n + 1) vertices) in T and by identifying opposite vertices on the boundary as indicated in Figure 8a A homeomorphic copy of the graph K is also embeddable in higher dimensional cubic lattices with 207 vertices (see [2]). ; see Figure 15.…”

“…The rest of the paper is organized as follows. In the next section we prove the existence of P 2 1 -graphs in planar lattices, and in the last section results about nonplanar lattices are discussed. Each section begins with a lemma, due to which the proofs of our other results are reduced to finding appropriate embeddings in the various lattices considered here.…”

“…In this paper, we are proving the existence of P 2 1 -graphs in various lattices. Moreover, it is worth mentioning that no P 3 1 -graph at all (whether planar or not) is known (see [14], p. 79).…”

“…Lemma The graph G is a P2 1 -graph if 2x ≥ y + 2z + 1 , u = 2x + 1 2 (y + 3t) + 4 , v = 2x − 2z + u, w = 9x + 2y + 3t + 17, and s = x − z + w .…”