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.

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

Graphs with Non-Concurrent Longest Paths

The most famous examples of Hypo-Hamiltonian graph is the Petersen graph. Before the discovery of Hypo-traceable graphs, Tibor Gallai, in 1966, raised the question whether the graphs in which each vertex is missed by some longest path. This property will be called Gallai’s property, various authors worked on that property. In 1969, Gallai’s question was first replied through H. Walther, who introduced a planar graph on 25 vertices satisfying Gallai’s criterion. Furthermore, H. Walther and H. Voss and Tudor Zamfirescu introduced the graph with 12 vertices and it was guessed that order 12 is the smaller possibility of such a graphLater the question was modifies by Tudor Zamfirescu and asked that whether there exists graphs of Paths and Cycles, that is to say i-connected graphs (planar or non-planar respectively), such that each set of j points are disjoint from some longest paths or cycles., Several good examples answering Tudor Zamfirescu’s questions were published. In this note a graphs is developed with the property that everyone vertex is missed by some longest cycle with connectivity 2, satisfying Gallai’s property.  The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

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Abdul Naeem Kalhoro

Papers

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

Graphs with Non-Concurrent Longest Paths

A Two-Connected Graph with Gallai’s Property

A Two-Connected Graph with Gallai’s Property