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


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.


Factors in graphs Cubic graphs Cut-edges 

Mathematics Subject Classification

05C07 05C70 05C35 



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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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