Independent systems of representatives in weighted graphs

Abstract

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s.

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References

  1. [1]

    R. Aharoni: Ryser’s conjecture for 3-partite 3-graphs, Combinatorica 21(1) (2001), 1–4.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    R. Aharoni, E. Berger and R. Ziv: A tree version of König’s theorem, Combinatorica 22(3) (2002), 335–343.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    R. Aharoni, M. Chudnovsky and A. Kotlov: Triangulated spheres and colored cliques, Disc. Comput. Geometry 28 (2002), 223–229.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    R. Aharoni and M. Chudnovsky: Special triangulations of the simplex and systems of disjoint representatives, unpublished.

  5. [5]

    R. Aharoni and P. Haxell: Hall’s theorem for hypergraphs, J. of Graph Theory 35 (2000), 83–88.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    A. Björner: Topological methods, in: Handbook of Combinatorics (R. Graham, M. Grötschel and L. Lovász editors), Elsevier and the MIT Press (1995).

  7. [7]

    M. Fellows: Transversals of vertex partitions in graphs, SIAM Journal of Disc. Math. 3 (1990), 206–215.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    H. Fleischner and M. Stiebitz: A solution to a coloring problem of P. Erdős, Discrete Math. 101 (1992), 39–48.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    F. Galvin: The list chromatic index of a bipartite multigraph, J. Combin. Theory Ser. B 63 (1995), 153–158.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    P. Hall: On representation of subsets, J. London Math. Soc. 10 (1935), 26–30.

    MATH  Article  Google Scholar 

  11. [11]

    P. E. Haxell: A condition for matchability in hypergraphs, Graphs and Combinatorics 11 (1995), 245–248.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    P. E. Haxell: A note on vertex list coloring, Combin. Probab. Comput. 10 (2001), 345–347.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    P. E. Haxell: On the strong chromatic number, Comb. Prob. and Computing 13 (2004), 857–865.

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

    P. E. Haxell: private communication.

  15. [15]

    R. Meshulam: The clique complex and hypergraph matching, Combinatorica 21(1) (2001), 89–94.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    R. Meshulam: Domination numbers and homology, J. Combin. Theory Ser. A 102 (2003), 321–330.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    R. Meshulam: private communication.

  18. [18]

    R. Yuster: Independent transversals in r-partite graphs, Discrete Math. 176 (1997), 255–261.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Ron Aharoni.

Additional information

The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.

The research of the third author was supported by the Sacta-Rashi Foundation.

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Aharoni, R., Berger, E. & Ziv, R. Independent systems of representatives in weighted graphs. Combinatorica 27, 253–267 (2007). https://doi.org/10.1007/s00493-007-2086-y

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Mathematics Subject Classification (2000)

  • 05C15