The asymptotic number of graphs not containing a fixed color-critical subgraph

Abstract

For a finite graphG letForb(H) denote the class of all finite graphs which do not containH as a (weak) subgraph. In this paper we characterize the class of those graphsH which have the property that almost all graphs inForb(H) are ℓ-colorable. We show that this class corresponds exactly to the class of graphs whose extremal graph is the Turán-graphT n (ℓ).An earlier result of Simonovits (Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions,Discrete Math. 7 (1974), 349–376) shows that these are exactly the (ℓ+1)-chromatic graphs which contain a color-critical edge.

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Prömel, H.J., Steger, A. The asymptotic number of graphs not containing a fixed color-critical subgraph. Combinatorica 12, 463–473 (1992). https://doi.org/10.1007/BF01305238

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