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Graphs and Combinatorics

, Volume 35, Issue 4, pp 805–813 | Cite as

Largest 2-Regular Subgraphs in 3-Regular Graphs

  • Ilkyoo Choi
  • Ringi Kim
  • Alexandr V. Kostochka
  • Boram Park
  • Douglas B. WestEmail author
Original Paper
  • 30 Downloads

Abstract

For a graph G, let \(f_2(G)\) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of \(f_2(G)\) over 3-regular n-vertex simple graphs G. To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most \(\max \{0,\lfloor (c-1)/2\rfloor \}\) vertices. More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most \(\max \{0,\lfloor (3n-2m+c-1)/2\rfloor \}\) vertices. These bounds are sharp; we describe the extremal multigraphs.

Keywords

Factors in graphs Cubic graphs Cut-edges 

Mathematics Subject Classification

05C07 05C70 05C35 

Notes

References

  1. 1.
    Edmonds, J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Nat. Bur. Stand. Sect. B 69B, 125–130 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Exoo, G., Jajcay, R.: Dynamic cage survey. Electron. J. Comb. 15, 4 (2008). Dynamic Survey #16 (updated 2013 to 55 pages)zbMATHGoogle Scholar
  4. 4.
    Gallai, T.: Neuer Beweis eines Tutteschen Satzes. Magyar Tud. Akad. Mat. Kut. Int. Közl 8, 135–139 (1963)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hanson, D., Loten, C.O.M., Toft, B.: On interval colourings of bi-regular bipartite graphs. Ars Comb. 50, 23–32 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Henning, M.A., Yeo, A.: Tight lower bounds on the size of a maximum matching in a regular graph. Graphs Comb. 23, 647–657 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kostochka, A.V., Raspaud, A., Toft, B., West, D.B., Zirlin, D.: Cut-edges and regular factors in regular graphs of odd degree. arXiv:1806.05347 (2018)
  8. 8.
    Naddef, D., Pulleyblank, W.R.: Matchings in regular graphs. Discrete Math. 34, 283–291 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    O, S., West, D.B.: Balloons, cut-edges, matchings, and total domination in regular graphs of odd degree. J. Graph Theory 64, 116–131 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    O, S., West, D.B.: Sharp bounds for the Chinese postman problem in 3-regular graphs and multigraphs. Discrete Appl. Math. 190–191, 163–168 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Petersen, J.: Die Theorie der regulären graphs. Acta Math. 15, 193–220 (1891)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Plesník, J.: Connectivity of regular graphs and the existence of 1-factors. Mat. Časopis Sloven. Akad. Vied 22, 310–318 (1972)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Tutte, W.T.: The factorization of linear graphs. J. Lond. Math. Soc. 22, 107–111 (1947)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • Ilkyoo Choi
    • 1
  • Ringi Kim
    • 2
  • Alexandr V. Kostochka
    • 3
    • 4
  • Boram Park
    • 5
  • Douglas B. West
    • 6
    • 7
    Email author
  1. 1.Department of MathematicsHankuk University of Foreign StudiesYongin-siRepublic of Korea
  2. 2.Department of Mathematical SciencesKAISTDaejeonRepublic of Korea
  3. 3.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  4. 4.Sobolev Institute of MathematicsNovosibirskRussia
  5. 5.Department of MathematicsAjou UniversitySuwon-siRepublic of Korea
  6. 6.Mathematics DepartmentZhejiang Normal UniversityJinhuaChina
  7. 7.Mathematics DepartmentUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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