A Planar Lattice Graph, with Empty Intersection of All Longest Path
Engineering Mathematics
Volume 3, Issue 1, June 2019, Pages: 6-8
Received: Mar. 5, 2019; Accepted: Jun. 12, 2019; Published: Jun. 26, 2019
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Authors
Abdul Naeem Kalhoro, Department of Mathematics, Faculty of Physical Sciences, Shah Abdul Latif University, Khairpur, Pakistan
Ali Dino Jumani, Department of Mathematics, Faculty of Physical Sciences, Shah Abdul Latif University, Khairpur, Pakistan
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Abstract
Tibor Gallai in 1966 elevated the declaration about the existence of graphs with the property that every vertex is missed by some longest path. This property will be called Gallai’s property. First answer back by H. Walther, who introduced a planar graph on 25 vertices satisfying Gallai’s property, and various authors worked on that property, after examples of such graphs were found while examining such n-dimensional Ln graphs with the property that every longest Paths have empty intersection, can be embeddable in IRn, Some in equilateral triangular lattice T, Square lattice L2, hexagonal lattice H, also on the torus, Mobius strip, and the Klein bottle but no hypo-Hamiltonian graphs are embeddable in the planar square lattice. In this paper we present a graph embeddable into Cubic lattices L3, such that graphs can also occur as sub graphs of the cubic lattices, and enjoying the property that every vertex is missed by some longest path. Here research has also significance in applications. What if several processing units are interlinked as parts of a lattice network. Some of them developing a chain of maximal length are used to solve a certain task. To get a self-stable fault-tolerant system, it is indispensable that in case of failure of any unit or link, another chain of same length, not containing the faulty unit or link, can exchange the chain originally used. This is exactly the case investigated here. We denote by Ln the n-dimensional cubic lattice in IRn.
Keywords
Longest Paths, Longest Cycles, Lattice Graphs, Cubic Lattices, Gallai's Property
To cite this article
Abdul Naeem Kalhoro, Ali Dino Jumani, A Planar Lattice Graph, with Empty Intersection of All Longest Path, Engineering Mathematics. Vol. 3, No. 1, 2019, pp. 6-8. doi: 10.11648/j.engmath.20190301.12
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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