Distance Graphs and T-Coloring

We discuss relationships amongT-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral setTthat contains 0, the asymptoticT-coloring ratioR(T) is equal to the fractional chromatic number of the distance graphG(Z,D), whereD=T?{0}. This fact is then used to study the distance graphs with distance sets of the formDm,k={1,2,?,m}?{k}. The chromatic numbers and the fractional chromatic numbers ofG(Z,Dm,k) are determined for all values ofmandk. Furthermore, circular chromatic numbers ofG(Z,Dm,k) for some special values ofmandkare obtained.

[1]  Imre Leader.  The fractional chromatic number of infinite graphs , 1995, J. Graph Theory.

[2]  Margit Voigt.  On the Chromatic Number of Distance Graphs , 1992, J. Inf. Process. Cybern..

[3]  Margit Voigt,et al.  Chromatic Number of Prime Distance Graphs , 1994, Discret. Appl. Math..

[4]  J. H. Rabinowitz,et al.  An Asymptotic Approach to the Channel Assignment Problem , 1985 .

[5]  Xuding Zhu,et al.  Distance graphs with missing multiples in the distance sets , 1999, J. Graph Theory.

[6]  Paul Erdös,et al.  Colouring the real line , 1985, J. Comb. Theory B.

[7]  Jerrold R. Griggs.  The Channel Assigment Problem for Mutually Adjacent Sites , 1994, J. Comb. Theory, Ser. A.

[8]  Kenneth Falconer,et al.  Unsolved Problems In Geometry , 1991 .

[9]  V. Klee,et al.  Combinatorial Geometry in the Plane , 1964 .

[10]  A. Vince,et al.  Star chromatic number , 1988, J. Graph Theory.

[11]  Basil Gordon,et al.  Sequences of Integers with Missing Differences , 1973, J. Comb. Theory A.

[12]  Gerard J. Chang,et al.  Integral distance graphs , 1997 .

[13]  Xuding Zhu,et al.  Circular Chromatic Numbers and Fractional Chromatic Numbers of Distance Graphs , 1998, Eur. J. Comb..

[14]  Daphne Der-Fen Liu.  T-colorings of graphs , 1992, Discret. Math..

[15]  Fred S. Roberts,et al.  T-colorings of graphs: recent results and open problems , 1991, Discret. Math..

[16]  Lingling Huang,et al.  Circular Chromatic Numbers of Distance Graphs with Distance Sets Missing Multiples , 2000, Eur. J. Comb..

[17]  W. K. Hale.  Frequency assignment: Theory and applications , 1980, Proceedings of the IEEE.

[18]  Arnfried Kemnitz,et al.  Coloring of integer distance graphs , 1998, Discret. Math..

[19]  Xuding Zhu.  Star chromatic numbers and products of graphs , 1992, J. Graph Theory.

[20]  Xuding Zhu,et al.  Pattern Periodic Coloring of Distance Graphs , 1998, J. Comb. Theory, Ser. B.

[21]  David C. Fisher,et al.  Fractional colorings with large denominators , 1995, J. Graph Theory.

[22]  Arundhati Raychaudhuri.  Further Results on T-Coloring and Frequency Assignment Problems , 1994, SIAM J. Discret. Math..

[23]  Daphne Der-Fen Liu.  T-Colorings and Chromatic Number of Distance Graphs , 2000, Ars Comb..

[24]  C. Berge.  Fractional Graph Theory , 1978 .

[25]  Soma Gupta,et al.  Sets of Integers with Missing Differences , 2000, J. Comb. Theory A.

[26]  Xuding Zhu,et al.  The chromatic numbers of distance graphs , 1997, Discret. Math..

[27]  Daphne Der-Fen Liu.  T-graphs and the channel assignment problem , 1996, Discret. Math..