Short Tours through Large Linear Forests

  • Uriel Feige
  • R. Ravi
  • Mohit Singh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected d-regular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His proof is based on van-der-Warden’s conjecture (proved independently by Egorychev (1981) and by Falikman (1981)) regarding the permanent of doubly stochastic matrices. We provide an exponential improvement in the rate of decrease of the o(1) term (thus increasing the range of d for which the upper bound on the tour length is nontrivial). Our proof does not use the van-der-Warden conjecture, and instead is related to the linear arboricity conjecture of Akiyama, Exoo and Harary (1981), or alternatively, to a conjecture of Magnant and Martin (2009) regarding the path cover number of regular graphs. More generally, for arbitrary connected graphs, our techniques provide an upper bound on the minimum tour length, expressed as a function of their maximum, average, and minimum degrees. Our bound is best possible up to a term that tends to 0 as the minimum degree grows.


Regular Graph Minimum Degree Parallel Edge Path Cover Tour Length 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Uriel Feige
    • 1
  • R. Ravi
    • 2
  • Mohit Singh
    • 3
  1. 1.Department of Computer ScienceThe Weizmann InstituteRehovotIsrael
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityUSA
  3. 3.Microsoft ResearchRedmondUSA

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