Abstract

The shortest cycle cover conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx1.619m$, and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx1.630m$.

MSC codes

  1. 05C21
  2. 05C38

Keywords

  1. cycle cover
  2. cycle double cover
  3. shortest cycle cover

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
N. Alon and M. Tarsi, Covering multigraphs by simple circuits, SIAM J. Algebraic Discrete Methods, 6 (1985), pp. 345–350.
2.
J.-C. Bermond, B. Jackson, and F. Jaeger, Shortest coverings of graphs with cycles, J. Combin. Theory Ser. B, 35 (1983), pp. 297–308.
3.
U. A. Celmins, On Cubic Graphs That Do Not Have an Edge 3-Coloring, Ph.D. thesis, University of Waterloo, Waterloo, Canada, 1984.
4.
R. Diestel, Graph Theory, 2nd ed., Grad. Texts in Math. 173, Springer-Verlag, New York, 2000.
5.
G. Fan, Covering weighted graphs by even subgraphs, J. Combin. Theory Ser. B, 49 (1990), pp. 137–141.
6.
G. Fan, Integer flows and cycle covers, J. Combin. Theory Ser. B, 54 (1992), pp. 113–122.
7.
G. Fan, Short cycle covers of cubic graphs, J. Graph Theory, 18 (1994), pp. 131–141.
8.
G. Fan and A. Raspaud, Fulkerson's conjecture and circuit covers, J. Combin. Theory Ser. B, 61 (1994), pp. 133–138.
9.
H. Fleischner, Eine gemeinsame Basis für die Theorie der Eulerschen Graphen und den Satz von Petersen, Monatsh. Math., 81 (1976), pp. 267–278.
10.
L. A. Goddyn, Cycle Covers of Graphs, Ph.D. thesis, University of Waterloo, Waterloo, Canada, 1988.
11.
B. Jackson, Shortest circuit covers and postman tours in graphs with a nowhere zero 4-flow, SIAM J. Comput., 19 (1990), pp. 659–665.
12.
B. Jackson, Shortest circuit covers of cubic graphs, J. Combin. Theory Ser. B, 60 (1994), pp. 299–307.
13.
F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B, 26 (1979), pp. 205–216.
14.
U. Jamshy and M. Tarsi, Short cycle covers and the cycle double cover conjecture, J. Combin. Theory Ser. B, 56 (1992), pp. 197–204.
15.
U. Jamshy, A. Raspaud, and M. Tarsi, Short circuit covers for regular matroids with nowhere zero 5-flow, J. Combin. Theory Ser. B, 43 (1987), pp. 354–357.
16.
T. Kaiser, D. Král', and S. Norine, Unions of Perfect Matchings in Cubic Graphs, in Topics in Discrete Mathematics, M. Klazar, J. Kratochvíl, J. Matoušek, R. Thomas, and P. Valtr, eds., Springer, Berlin, Heidelberg, New York, 2006, pp. 225–230.
17.
D. Král', E. Máčajová, O. Pangrác, A. Raspaud, J.-S. Sereni, and M. Škoviera, Projective, affine, and Abelian colorings of cubic graphs, European J. Combin., 30 (2009), pp. 53–69.
18.
L. Lovász, On some connectivity properties of Eulerian graphs, Acta Math. Acad. Sci. Hungar., 28 (1976), pp. 129–138.
19.
E. Máčajová and M. Škoviera, Fano colourings of cubic graphs and the Fulkerson conjecture, Theoret. Comput. Sci., 349 (2005), pp. 112–120.
20.
W. Mader, A reduction method for edge-connectivity in graphs, Ann. Discrete Math., 3 (1978), pp. 145–164.
21.
M. Preissmann, Sur les colorations des arêtes des graphes cubiques, Thèse de Doctorat de 3eme, Universite of Grenoble, Grenoble, France, 1981.
22.
A. Raspaud, Cycle covers of graphs with a nowhere-zero 4-flow, J. Graph Theory, 15 (1991), pp. 649–654.
23.
P. D. Seymour, Sums of Circuits, in Graph Theory and Related Topics, J. A. Bondy and U. S. R. Murty, eds., Academic Press, New York, London, 1979, pp. 341–355.
24.
G. Szekeres, Polyhedral decompositions of cubic graphs, Bull. Austral. Math. Soc., 8 (1973), pp. 367–387.
25.
Z. Szigeti, Edge-connectivity augmentation of graphs over symmetric parity families, Discrete Math., 308 (2008), pp. 6527–6532.
26.
C. Q. Zhang, Integer Flows and Cycle Covers of Graphs, Marcel Dekker, Inc., New York, 1997.
27.
C. Q. Zhang, Circular flows of nearly Eulerian graphs and vertex-splitting, J. Graph Theory, 40 (2002), pp. 147–161.

Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 330 - 355
ISSN (online): 1095-7146

History

Submitted: 3 March 2008
Accepted: 8 February 2010
Published online: 9 April 2010

MSC codes

  1. 05C21
  2. 05C38

Keywords

  1. cycle cover
  2. cycle double cover
  3. shortest cycle cover

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Figures

Tables

Media

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media