# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## 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

 [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. [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. [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. [7] D.B. West, Introduction to Graph Theory (Prentice Hall, Upper Saddle River, NJ, 1996). [8] D. West, in: [1], Feb. 20, 2001.