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

Abdul Naeem Kalhoro,Ali Dino Jumani

doi:10.11648/j.engmath.20190301.12

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.

Highlights

A graph is Hamiltonian if there exists a Hamiltonian cycle in, i.e. a cycle which passes through every vertex of graph
Especially in graph theory, and was a lifelong friend and collaborator of Paul Erdős. He was a student of Dénes Kőnig and an advisor of László Lovász. He was a corresponding member of the Hungarian Academy of Sciences) asked whether there exist connected graphs with the property that every vertex is missed by some longest path
Zamfirescu’s questions about the existence of connected graphs with the property that for any vertices there is a longest path avoiding all of them, it was asked about examples with higher connectivity [3]

Summary

Introduction

A graph is Hamiltonian if there exists a Hamiltonian cycle in , i.e. a cycle which passes through every vertex of graph. Zamfirescu’s questions about the existence of (small, if possible minimal) connected graphs with the property that for any vertices there is a longest path avoiding all of them, it was asked about examples with higher connectivity [3]. Such graphs have been subsequently found, up to connectivity number 3 [4, 5, 6]. Shabbir [11] introduced connected graphs such that every pair of vertices is missed by a longest path in the triangular, square, and hexagonal lattices. They found such graphs in some lattices embedded on the torus, Mobius strip, and the Klein bottle

Embeddings of Graph with Empty Intersection of All Longest Paths

Conclusion