Advertisement

Combinatorica

, Volume 39, Issue 1, pp 91–133 | Cite as

A Unified Erdős–Pósa Theorem for Constrained Cycles

  • Tony HuynhEmail author
  • Felix Joos
  • Paul Wollan
Article
  • 47 Downloads

Abstract

A (Γ1,Γ2)-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups Γ1,Γ2. A cycle in such a labeled graph is (Γ1,Γ2)-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to (Γ1,Γ2)-labeled graphs. As an application, we determine all canonical obstructions to the Erdős–Pósa property for (Γ1,Γ2)-non-zero cycles in (Γ1,Γ2)-labeled graphs. The obstructions imply that the half-integral Erdős–Pósa property always holds for (Γ1,Γ2)-non-zero cycles.

Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erdős–Pósa property for cycles and S-cycles and the half-integral Erdős–Pósa property for odd cycles and odd S-cycles. Furthermore, we recover Reed's Escher-wall Theorem.

We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erdős–Pósa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and S-cycles not homologous to zero. Moreover, the (full) Erdős–Pósa property holds for S1-S2-cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erdős–Pósa property for cycles not homologous to zero and for odd S-cycles.

Mathematics Subject Classification (2000)

05C70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Birmelé, J.A. Bondy and B. Reed: The Erdős–Pósa property for long circuits, Combinatorica 27 (2007), 135–145.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    H. Bruhn, F. Joos and O. Schaudt: Long cycles through prescribed vertices have the Erdős–Pósa property, to appear in J. Graph Theory.Google Scholar
  3. [3]
    M. Chudnovsky, J. Geelen, B. Gerards, L. Goddyn, M. Lohman and P. Seymour: Non-zero A-paths in group-labelled graphs, Combinatorica 26 (2006), 521–532.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. P. Dilworth: A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51 (1950), 161–166.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. Erdős and L. Pósa: On the maximal number of disjoint circuits of a graph, Publ. Math. Debrecen 9 (1962), 3–12.MathSciNetzbMATHGoogle Scholar
  6. [6]
    P. Erdős and G. Szekeres: A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.MathSciNetzbMATHGoogle Scholar
  7. [7]
    S. Fiorini and A. Herinckx: A tighter Erdős–Pósa function for long cycles, J. Graph Theory 77 (2013), 111–116.CrossRefzbMATHGoogle Scholar
  8. [8]
    J. Geelen and B. Gerards: Excluding a group-labelled graph, J. Combin. Theory Ser. B 99 (2009), 247–253.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Geelen, B. Gerards, B. Reed, P. Seymour and A. Vetta: On the odd-minor variant of Hadwiger's conjecture, J. Combin. Theory (Series B) 99 (2009), 20–29.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    T. Huynh: The linkage problem for group-labelled graphs, PhD thesis, University of Waterloo, 2009.Google Scholar
  11. [11]
    N. Kakimura, K. Kawarabayashi and D. Marx: Packing cycles through prescribed vertices, J. Combin. Theory Ser. B 101 (2011), 378–381.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    K. Kawarabayashi and N. Kakimura: Half-integral packing of odd cycles through prescribed vertices, Combinatorica 33 (2014), 549–572.MathSciNetzbMATHGoogle Scholar
  13. [13]
    K. Kawarabayashi, R. Thomas and P. Wollan: A new proof of the Flat Wall Theorem, J. Combin. Theory Ser. B 129 (2018), 204–238.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    D. Lokshtanov, M. S. Ramanujan and S. Saurabh: The half-integral Erdős–Pósa property for non-null cycles, arXiv:1703.02866, 2017.Google Scholar
  15. [15]
    M. Pontecorvi and P. Wollan: Disjoint cycles intersecting a set of vertices, J. Combin. Theory (Series B) 102 (2012), 1134–1141.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    B. Reed: Mangoes and blueberries, Combinatorica 19 (1999), 267–296.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    N. Robertson and P. Seymour: Graph minors. X. Obstructions to tree-decomposition, J. Combin. Theory Ser. B 52 (1991), 153–190.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N. Robertson and P. Seymour: Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B 63 (1995), 65–110.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Thomassen: On the presence of disjoint subgraphs of a specified type, J. Graph Theory 12 (1988), 101–111.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    P. Wollan: Packing cycles with modularity constraints, Combinatorica 31 (2011), 95–126.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    T. Zaslavsky: Biased graphs. I. Bias, balance, and gains, J. Combin. Theory Ser. B 47 (1989), 32–52.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    T. Zaslavsky: Biased graphs. II. The three matroids, J. Combin. Theory Ser. B 51 (1991), 46–72.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBruxellesBelgium
  2. 2.Institut für Optimierung und Operations ResearchUniversität UlmUlmGermany
  3. 3.Department of Computer ScienceUniversity of RomeRomeItaly

Personalised recommendations