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Amalgamations and Equitable Block-Colorings

  • E. B. Matson
  • C. A. Rodger
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 834)

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 (sp)-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 (sp)-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.

References

  1. 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.20617MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buchanan, H.: Graph factors and hamiltonian decompositions. Ph.D. Dissertation. University of West Virginia (1998)Google Scholar
  3. 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.2015.01.035MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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/s00373-015-1648-9MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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.2009.06.032MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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/s00373-008-0794-8MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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.2010.03.018MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)90020-0CrossRefzbMATHGoogle Scholar
  9. 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-XCrossRefzbMATHGoogle Scholar
  10. 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.1024CrossRefzbMATHGoogle Scholar
  11. 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.2001.2104MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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.10142MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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.2015.12.029CrossRefzbMATHGoogle Scholar
  14. 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. 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.2017.10.018CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

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