# Decompositions into two paths

Discussiones Mathematicae Graph Theory (2005)

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

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topZdzisław 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

top- [1] http://listserv.nodak.edu/archives/graphnet.html.
- [2] S. Lin, Computer solutions of the traveling salesman problem, Bell System Tech. J. 44 (1965) 2245-2269. Zbl0136.14705
- [3] Z. Skupień, Sparse hamiltonian 2-decompositions together with numerous Hamilton cycles, submitted.
- [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] K.W. Smith, Two-path conjecture, in: [1], Feb. 16, 2001.
- [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] D.B. West, Introduction to Graph Theory (Prentice Hall, Upper Saddle River, NJ, 1996). Zbl0845.05001
- [8] D. West, in: [1], Feb. 20, 2001.

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