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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1285,
author = {Zdzis\l aw Skupie\'n},
title = {Decompositions into two paths},
journal = {Discussiones Mathematicae Graph Theory},
volume = {25},
year = {2005},
pages = {325-329},
zbl = {1103.05069},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1285}
}
Zdzisław Skupień. Decompositions into two paths. Discussiones Mathematicae Graph Theory, Tome 25 (2005) pp. 325-329. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1285/
[000] [1] http://listserv.nodak.edu/archives/graphnet.html.
[001] [2] S. Lin, Computer solutions of the traveling salesman problem, Bell System Tech. J. 44 (1965) 2245-2269. | Zbl 0136.14705
[002] [3] Z. Skupień, Sparse hamiltonian 2-decompositions together with numerous Hamilton cycles, submitted.
[003] [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. | Zbl 0176.22402
[004] [5] K.W. Smith, Two-path conjecture, in: [1], Feb. 16, 2001.
[005] [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. | Zbl 0382.05039
[006] [7] D.B. West, Introduction to Graph Theory (Prentice Hall, Upper Saddle River, NJ, 1996). | Zbl 0845.05001
[007] [8] D. West, in: [1], Feb. 20, 2001.