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Hamiltonicity of a class of toroidal graphs

  • Dipendu Maity EMAIL logo and Ashish Kumar Upadhyay
Published/Copyright: March 10, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 2

Abstract

If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

MSC 2010: Primary 52B70; 05C10; 05C45
Keywords: Semi-equivelar Maps; Hamiltonian cycles; connectivity

  1. Communicated by Peter Horák

Acknowledgement

AKU acknowledges financial support from DST-SERB grant number SR/S4/MS:717/10. The authors thank Prof. M. Hornak for numerous useful comments, substantial improvement of proofs and for drawing their attention to the references [2] and [3].

References

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Received: 2018-06-09
Accepted: 2019-05-12
Published Online: 2020-03-10
Published in Print: 2020-04-28

© 2020 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0367
Keywords for this article
Semi-equivelar Maps; Hamiltonian cycles; connectivity

Articles in the same Issue

  1. Regular papers
  2. Relatively residuated lattices and posets
  3. Strongly s-dense injective hull and Banaschewski’s theorems for acts
  4. Wild sets in global function fields
  5. Generators and integral points on elliptic curves associated with simplest quartic fields
  6. New Filbert and Lilbert matrices with asymmetric entries
  7. Returning functions with closed graph are continuous
  8. On sets of points of approximate continuity and ϱ-upper continuity
  9. Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
  10. On solvability of some nonlocal boundary value problems for biharmonic equation
  11. The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
  12. Strongly increasing solutions of higher-order quasilinear ordinary differential equations
  13. Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
  14. Filippov solutions of vector Dirichlet problems
  15. Sequences of positive homoclinic solutions to difference equations with variable exponent
  16. Modified Lupaş-Jain operators
  17. New fixed point results in bv(s)-metric spaces
  18. Improved Young and Heinz operator inequalities for unitarily invariant norms
  19. Scrutiny of some fixed point results by S-operators without triangular inequality
  20. The lattices of families of regular sets in topological spaces
  21. Iterated partial summations applied to finite-support discrete distributions
  22. Hamiltonicity of a class of toroidal graphs
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