Local chromatic number and Sperner capacity

We introduce a directed analog of the local chromatic number defined by Erdos et al. [Discrete Math. 59 (1986) 21-34] and show that it provides an upper bound for the Sperner capacity of a directed graph. Applications and variants of this result are presented. In particular, we find a special orientation of an odd cycle and show that it achieves the maximum of Sperner capacity among the differently oriented versions of the cycle. We show that apart from this orientation, for all the others an odd cycle has the same Sperner capacity as a single edge graph. We also show that the (undirected) local chromatic number is bounded from below by the fractional chromatic number while for power graphs the two invariants have the same exponential asymptotics (under the co-normal product on which the definition of Sperner capacity is based). We strengthen our bound on Sperner capacity by introducing a fractional relaxation of our directed variant of the local chromatic number.

[1]  Miklós Simonovits,et al.  The coloring numbers of the direct product of two hypergraphs , 1974 .

[2]  Noga Alon.  On the Capacity of Digraphs , 1998, Eur. J. Comb..

[3]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[4]  E. Capone.  Hide and Seek , 1991 .

[5]  Gábor Tardos,et al.  Local Chromatic Number, KY Fan's Theorem, And Circular Colorings , 2006, Comb..

[6]  János Körner,et al.  Colour Number, Capacity, and Perfectness of Directed Graphs , 2000, Graphs Comb..

[7]  János Körner,et al.  Different capacities of a digraph , 1994, Graphs Comb..

[8]  L. A. Shepp,et al.  The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle , 1993 .

[9]  P. Frankl,et al.  The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle , 1993, Proceedings. IEEE International Symposium on Information Theory.

[10]  Henry Meyniel,et al.  On the Shannon Capacity of a Directed Graph , 1985, Eur. J. Comb..

[11]  Gábor Simonyi,et al.  Orientations of Self-complementary Graphs and the Relation of Sperner and Shannon Capacities , 1999, Eur. J. Comb..

[12]  Luisa Gargano,et al.  Capacities: From Information Theory to Extremal Set Theory , 1994, J. Comb. Theory, Ser. A.

[13]  Noga Alon,et al.  The Shannon Capacity of a Union , 1998, Comb..

[14]  Alon Orlitsky,et al.  Zero-Error Information Theory , 1998, IEEE Trans. Inf. Theory.

[15]  A. Blokhuis.  On the Sperner Capacity of the Cyclic Triangle , 1993 .

[16]  R. McEliece,et al.  Hide and Seek, Data Storage, and Entropy , 1971 .

[17]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[18]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[19]  Noga Alon,et al.  Graph Powers , 2002 .

[20]  Vojtech Rödl,et al.  Coloring graphs with locally few colors , 1986, Discret. Math..

[21]  Dennis Fregger,et al.  Colour by number , 2007 .

[22]  Luisa Gargano,et al.  Qualitative Independence and Sperner Problems for Directed Graphs , 1992, J. Comb. Theory, Ser. A.

[23]  Ron Holzman,et al.  A nontrivial lower bound on the Shannon capacities of the complements of odd cycles , 2003, IEEE Trans. Inf. Theory.

[24]  Gábor Simonyi,et al.  A Sperner-Type Theorem and Qualitative Independence , 1992, J. Comb. Theory, Ser. A.