Skip to main content
SpringerLink
Log in

Amalgamations and Equitable Block-Colorings

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 834))

Included in the following conference series:

  • 595 Accesses

Abstract

An H-decomposition of G is a partition P of E(G) into blocks, each element of which induces a copy of H. Amalgamations of graphs have proved to be a valuable tool in the construction of H-decompositions. The method can force decompositions to satisfy fairness notions. Here the use of the method is further applied to (s, p)-equitable block-colorings of H-decompostions: a coloring of the blocks using exactly s colors such that each vertex v is incident with blocks colored with exactly p colors, the blocks containing v being shared out as evenly as possible among the p color classes. Recently interest has turned to the color vector \(V(E)=(c_1(E), c_2(E),\) \(\dots , c_s(E))\) of such colorings. Amalgamations are used to construct (s, p)-equitable block-colorings of \(C_4\)-decompositions of \(K_n - F\) and \(K_2\)-decompositions of \(K_{n}\), focusing on one unsolved case with each where \(c_1\) is as small as possible and \(c_2\) is as large as possible.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bahmanian, M.A., Rodger, C.A.: Multiply balanced edge-colorings of multigraphs. J. Graph Theory 70, 297–317 (2012). https://doi.org/10.1002/jgt.20617

    Article  MathSciNet  MATH  Google Scholar 

  2. Buchanan, H.: Graph factors and hamiltonian decompositions. Ph.D. Dissertation. University of West Virginia (1998)

    Google Scholar 

  3. Erzurumluoğlu, A., Rodger, C.A.: Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames. Discret. Math. 338, 1173–1177 (2015). https://doi.org/10.1016/j.disc.2015.01.035

    Article  MathSciNet  MATH  Google Scholar 

  4. Erzurumluoğlu, A., Rodger, C.A.: Fair 1-factorizations and fair holey 1-factorizations of complete multipartite graphs. Graphs Comb. 32, 1377–1388 (2016). https://doi.org/10.1007/s00373-015-1648-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Gionfriddo, L., Gionfriddo, M., Ragusa, G.: Equitable specialized block-colourings for 4-cycle systems-I. Discret. Math. 310, 3126–3131 (2010). https://doi.org/10.1016/j.disc.2009.06.032

    Article  MathSciNet  MATH  Google Scholar 

  6. Gionfriddo, M., Hork, P., Milazzo, L., Rosa, A.: Equitable specialized block-colourings for Steiner triple systems. Graphs Combin. 24, 313–326 (2008). https://doi.org/10.1007/s00373-008-0794-8

    Article  MathSciNet  MATH  Google Scholar 

  7. Gionfriddo, M., Ragusa, G.: Equitable specialized block-colourings for 4-cycle systems-II. Discret. Math. 310(13–14), 1986–1994 (2010). https://doi.org/10.1016/j.disc.2010.03.018

    Article  MathSciNet  MATH  Google Scholar 

  8. Hilton, A.J.W.: Hamilton decompositions of complete graphs. J. Comb. Theory Ser. B 36, 125–134 (1984). https://doi.org/10.1016/0095-8956(84)90020-0

    Article  MATH  Google Scholar 

  9. Hilton, A.J.W., Rodger, C.A.: Hamilton decompositions of complete regular \(s\)-partite graphs. Discret. Math. 58, 63–78 (1986). https://doi.org/10.1016/0012-365X(86)90186-X

    Article  MATH  Google Scholar 

  10. Leach, C.D., Rodger, C.A.: Non-disconnecting disentanglements of amalgamated \(2\)-factorizations of complete multipartite graphs. J. Comb. Des. 9, 460–467 (2001). https://doi.org/10.1002/jcd.1024

    Article  MATH  Google Scholar 

  11. Leach, C.D., Rodger, C.A.: Fair hamilton decompositions of complete multipartite graphs. J. Comb. Theory Ser. B 85, 290–296 (2002). https://doi.org/10.1006/jctb.2001.2104

    Article  MathSciNet  MATH  Google Scholar 

  12. Leach, C.D., Rodger, C.A.: Hamilton decompositions of complete multipartite graphs with any \(2\)-factor leave. J. Graph Theory 44, 208–214 (2003). https://doi.org/10.1002/jgt.10142

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, S., Rodger, C.A.: Equitable block-colorings of \(C_4\)-deompostions of \(K_v-F\). Discret. Math. 339, 1519–1524 (2016). https://doi.org/10.1016/j.disc.2015.12.029

    Article  MATH  Google Scholar 

  14. Li, S., Matson, E.B., Rodger, C.A.: Extreme equitable block-colorings of \(C_4\)-decompositions of \(K_v-F\). Australas. J. Comb. 71(1), 92–103 (2018)

    Google Scholar 

  15. Matson, E.B., Rodger, C.A.: More extreme equitable colorings of decompositions of \(K_v\) and \(K_v-F\). Discret. Math. 341, 1178–1184 (2018). https://doi.org/10.1016/j.disc.2017.10.018

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL, 36849, USA

    E. B. Matson & C. A. Rodger

Authors
  1. E. B. Matson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. A. Rodger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. B. Matson .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology BHU, Varanasi, Uttar Pradesh, India

    Debdas Ghosh

  2. Haldia Institute of Technology, Haldia, India

    Debasis Giri

  3. University of Central Florida, Orlando, Florida, USA

    Ram N. Mohapatra

  4. Istanbul Commerce University, Istanbul, Turkey

    Ekrem Savas

  5. Kyushu University, Fukuoka, Japan

    Kouichi Sakurai

  6. Indian Institute of Technology (BHU), Varanasi, India

    L. P. Singh

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Matson, E.B., Rodger, C.A. (2018). Amalgamations and Equitable Block-Colorings. In: Ghosh, D., Giri, D., Mohapatra, R., Savas, E., Sakurai, K., Singh, L. (eds) Mathematics and Computing. ICMC 2018. Communications in Computer and Information Science, vol 834. Springer, Singapore. https://doi.org/10.1007/978-981-13-0023-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-981-13-0023-3_5

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-13-0022-6

  • Online ISBN: 978-981-13-0023-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation