Decompositions into two paths

Zdzisław Skupień

Discussiones Mathematicae Graph Theory (2005)

  • Volume: 25, Issue: 3, page 325-329
  • ISSN: 2083-5892

Abstract

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It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The Thomason result on hamiltonian pairs is used and is proved to be sharp.

How to cite

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Zdzisław Skupień. "Decompositions into two paths." Discussiones Mathematicae Graph Theory 25.3 (2005): 325-329. <http://eudml.org/doc/270615>.

@article{ZdzisławSkupień2005,
abstract = {It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The Thomason result on hamiltonian pairs is used and is proved to be sharp.},
author = {Zdzisław Skupień},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph; multigraph; path decomposition; hamiltonian decomposition; traceable; connected multigraph; traceable pairs},
language = {eng},
number = {3},
pages = {325-329},
title = {Decompositions into two paths},
url = {http://eudml.org/doc/270615},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Zdzisław Skupień
TI - Decompositions into two paths
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 3
SP - 325
EP - 329
AB - It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The Thomason result on hamiltonian pairs is used and is proved to be sharp.
LA - eng
KW - graph; multigraph; path decomposition; hamiltonian decomposition; traceable; connected multigraph; traceable pairs
UR - http://eudml.org/doc/270615
ER -

References

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  1. [1] http://listserv.nodak.edu/archives/graphnet.html. 
  2. [2] S. Lin, Computer solutions of the traveling salesman problem, Bell System Tech. J. 44 (1965) 2245-2269. Zbl0136.14705
  3. [3] Z. Skupień, Sparse hamiltonian 2-decompositions together with numerous Hamilton cycles, submitted. 
  4. [4] N.J.A. Sloane, Hamiltonian cycles in a graph of degree four, J. Combin. Theory 6 (1969) 311-312, doi: 10.1016/S0021-9800(69)80093-1. Zbl0176.22402
  5. [5] K.W. Smith, Two-path conjecture, in: [1], Feb. 16, 2001. 
  6. [6] A.G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs, in: B. Bollobás, ed., Advances in Graph Theory (Proc. Cambridge Combin. Conf., 1977), Ann. Discrete Math. 3 (1978) (North-Holland, Amsterdam, 1978) pp. 259-268. Zbl0382.05039
  7. [7] D.B. West, Introduction to Graph Theory (Prentice Hall, Upper Saddle River, NJ, 1996). Zbl0845.05001
  8. [8] D. West, in: [1], Feb. 20, 2001. 

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