Independent systems of representatives in weighted graphs Authors Ron Aharoni Department of Mathematics Technion Eli Berger Department of Mathematics Technion Department of Mathematics Princeton University Ran Ziv Department of Computer Science Tel-Hai Academic College Article

First Online: 29 May 2008 Received: 15 June 2003 Revised: 03 January 2005 DOI :
10.1007/s00493-007-2086-y

Cite this article as: Aharoni, R., Berger, E. & Ziv, R. Combinatorica (2007) 27: 253. doi:10.1007/s00493-007-2086-y
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.

Mathematics Subject Classification (2000) 05C15 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.

References [1]

R. Aharoni : Ryser’s conjecture for 3-partite 3-graphs,

Combinatorica
21(1) (2001), 1–4.

MATH CrossRef MathSciNet [2]

R. Aharoni, E. Berger and

R. Ziv : A tree version of König’s theorem,

Combinatorica
22(3) (2002), 335–343.

MATH CrossRef MathSciNet [3]

R. Aharoni, M. Chudnovsky and

A. Kotlov : Triangulated spheres and colored cliques,

Disc. Comput. Geometry
28 (2002), 223–229.

MATH CrossRef MathSciNet [4]

R. Aharoni and M. Chudnovsky : Special triangulations of the simplex and systems of disjoint representatives, unpublished .

[5]

R. Aharoni and

P. Haxell : Hall’s theorem for hypergraphs,

J. of Graph Theory
35 (2000), 83–88.

MATH CrossRef MathSciNet [6]

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

[7]

M. Fellows : Transversals of vertex partitions in graphs,

SIAM Journal of Disc. Math.
3 (1990), 206–215.

MATH CrossRef MathSciNet [8]

H. Fleischner and

M. Stiebitz : A solution to a coloring problem of P. Erdős,

Discrete Math.
101 (1992), 39–48.

MATH CrossRef MathSciNet [9]

F. Galvin : The list chromatic index of a bipartite multigraph,

J. Combin. Theory Ser. B
63 (1995), 153–158.

MATH CrossRef MathSciNet [10]

P. Hall : On representation of subsets,

J. London Math. Soc.
10 (1935), 26–30.

MATH CrossRef [11]

P. E. Haxell : A condition for matchability in hypergraphs,

Graphs and Combinatorics
11 (1995), 245–248.

MATH CrossRef MathSciNet [12]

P. E. Haxell : A note on vertex list coloring,

Combin. Probab. Comput.
10 (2001), 345–347.

MATH CrossRef MathSciNet [13]

P. E. Haxell : On the strong chromatic number,

Comb. Prob. and Computing
13 (2004), 857–865.

MATH CrossRef MathSciNet [14]

P. E. Haxell : private communication .

[15]

R. Meshulam : The clique complex and hypergraph matching,

Combinatorica
21(1) (2001), 89–94.

MATH CrossRef MathSciNet [16]

R. Meshulam : Domination numbers and homology,

J. Combin. Theory Ser. A
102 (2003), 321–330.

MATH CrossRef MathSciNet [17]

R. Meshulam : private communication .

[18]

R. Yuster : Independent transversals in

r -partite graphs,

Discrete Math.
176 (1997), 255–261.

MATH CrossRef MathSciNet