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Discussiones Mathematicae Graph Theory 25(3) (2005) 325-329
DOI: https://doi.org/10.7151/dmgt.1285

DECOMPOSITIONS INTO TWO PATHS

Zdzisław Skupień

Faculty of Applied Mathematics
AGH University of Science and Technology
al. Mickiewicza 30, 30-059 Kraków, Poland
e-mail: skupien@uci.agh.edu.pl

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.

Keywords: graph, multigraph, path decomposition, hamiltonian decomposition, traceable.

2000 Mathematics Subject Classification: 05C70, 05C35, 05C38, 05C45.

References

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Received 8 March 2004
Revised 2 November 2004