Skip to main content
Springer Nature Link
Log in

Circuit Decompositions and Shortest Circuit Coverings of Hypergraphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

It is one of fundamental theorems in graph theory that every even graph has a circuit decomposition. This classical result for ordinary graphs is extended in this paper for uniform bridgeless hypergraphs if the degree of every vertex is even. One of major open problems for shortest circuit cover was a conjecture proposed by Itai and Rodeh (Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 62, pp. 289–299. Springer, Berlin, 1978) that every bridgeless graph G has a circuit cover of total length at most \(|E|+|V|-1\). This conjecture was solved by Fan (J Combin Theory Ser B 74:353–367, 1998) for ordinary graphs, and is extended in this paper for bridgeless hypergraphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Alon, N., Tarsi, M.: Covering multigraphs by simple circuits. SIAM J. Algebraic Discrete Methods 6, 345–350 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bahmanian, M.A., Šajna, M.: Quasi-Eulerian Hypergraphs. Electron. J. Comb. 24(3), 3–30 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam, London (1973)

  4. Bermond, J.C., Jackson, B., Jaeger, F.: Shortest covering of graphs with cycles. J. Combin. Theory Ser. B 35, 297–308 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bollobas, B., Saito, A., Wormald, N.C.: Regular factors of regular graphs. J. Graph Theory 9, 97–103 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)

    Book  MATH  Google Scholar 

  7. Fan, G.-H.: Integer flows and cycle covers. J. Combin. Theory Ser. B 54, 113–122 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan, G.-H.: Covering graphs by cycles. SIAM J. Discrete Math. 5, 491–496 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan, G.-H.: Short cycle covers of cubic graphs. J. Graph Theory 18, 131–141 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, G.-H., Raspaud, A.: Fulkerson’s conjecture and circuits covers. J. Combin. Theory Ser. B 61, 133–138 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fan, G.-H.: Proofs of two minimum circuit cover conjectures. J. Combin. Theory Ser. B 74, 353–367 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fleischner, H.: Eine gemeinsame Basis für die Theorie der eulerschen Graphen und den Satz von Petersen. Monatsh. Math. 81, 267–278 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Itai, A., Rodeh, M.: Covering a graph by circuits. In: “Automata, Languages and Programming,” Lecture Notes in Computer Science, Vol. 62, pp. 289-299. Springer, Berlin (1978)

  14. Jaeger, F.: Flows and generalized coloring theorems in graphs. J. Combin. Theory Ser. B 26, 205–216 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jamshy, U., Raspaud, A., Tarsi, M.: Short circuit covers for regular matroids with nowhere-zero \(5\)-flow. J. Combin. Theory Ser. B 43, 354–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lonc, Z., Naroski, P.: On tours that contain all edges of a hypergraph. Electron. J. Comb. 17(R144), 1 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Máčajová, E., Raspaud, A., Tarsi, M., Zhu, X.-D.: Short cycle covers of graphs and nowhere-zero flows. J. Graph Theory 68, 340–348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Máčajová, E., Raspaud, A., Rollová, E., Škoviera, M.: Circuit covers of signed graphs. J. Graph Theory 81, 120–133 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Petersen, J.: Die Theorie der regulären Graphs. Acta Math. 15, 193–220 (1891)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sajna, M.: Eulerian-type properties of hypregraphs (slides) presented at Contributed Minisymposia “Hypergraphs(CM22)”. CanaDAM 2013—Canadian Mathematical Society (2013)

  21. Seymour, P.D.: On multi-colorings of cubic graphs and the conjecture of Fulkerson and Tutte. Proc. Lond. Math. Soc. s3–38, 423–460 (1979)

    Article  MATH  Google Scholar 

  22. Szekeres, G.: Polyhedral decompositions of cubic graphs. Bull. Austr. Math. Soc. 8, 367–387 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tutte, W.T.: Personal correspondence with H. Fleischner (July 22nd, 1987)

  24. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

  25. Zhang, C.-Q.: Circular flows of nearly eulerian graphs and vertex splitting. J. Graph Theory 40, 147–161 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Liying Kang

  2. College of Arts and Sciences, Shanghai Maritime University, Shanghai, 201306, People’s Republic of China

    Weihua Lu

  3. Ocean College, Zhejiang University, Zhoushan, 316021, Zhejiang, People’s Republic of China

    Yezhou Wu

  4. Department of Mathematical Sciences and Center for Computational Sciences, Middle Tennessee State University, Murfreesboro, TN, 37132, USA

    Dong Ye

  5. Department of Mathematics, West Virginia University, Morgantown, WV, 26506, USA

    Cun-Quan Zhang

Authors
  1. Liying Kang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Weihua Lu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yezhou Wu
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Dong Ye
    View author publications

    You can also search for this author inPubMed Google Scholar

  5. Cun-Quan Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Cun-Quan Zhang.

Additional information

Liying Kang: Partially supported by National Natural Science Foundation of China (Nos. 11471210 and 11571222). Yezhou Wu: Partially supported by National Natural Science Foundation of China (No. 11501504) and Natural Science Foundation of Zhejiang Province (No. LY16A010005). Dong Ye: Partially supported by Simons Foundation (No. 396516). Cun-Quan Zhang: Partially supported by National Security Agency (No. H98230-16-1-0004) and National Science Foundation (No. DMS-1700218).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kang, L., Lu, W., Wu, Y. et al. Circuit Decompositions and Shortest Circuit Coverings of Hypergraphs. Graphs and Combinatorics 34, 365–372 (2018). https://doi.org/10.1007/s00373-018-1881-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-018-1881-0

Keywords