A Planar Lattice Graph, with Empty Intersection of All Longest Path

Abdul Naeem Kalhoro, Ali Dino Jumani
2019 Engineering Mathematics  
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 L n graphs with the property that every longest Paths have empty intersection,
more » ... can be embeddable in IR n , Some in equilateral triangular lattice T, Square lattice L 2 , 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 L 3 , 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 L n the n-dimensional cubic lattice in IR n .
doi:10.11648/j.engmath.20190301.12 fatcat:nqb3sqzicvazbh4di42wu6sgtq