Fractional chromatic number of a random subgraph

Abstract

It is well known [6] that a random subgraph of the complete graph K n has chromatic number Θ(n/ log n) w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any n-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least n, the fractional chromatic number of its random subgraph is at least n/(8 log 2 (4n)) with probability more than 1 − 1 2n . This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.

On the Chromatic Number in the Stochastic Block Model

“…Determining the chromatic number of a random subgraph G p for a general graph G is a much harder problem; see, for example, [4][5][6]29,34]. In particular, Bukh asks [6] whether for any graph G, there exists a positive constant c such that Eχ(G 1/2 ) c log(χ(G)) χ(G).…”

Expected Chromatic Number of Random Subgraphs

“…where α(H) denotes the independence number of H (i.e., the maximum size of a set of vertices containing no edges), and Mohar [14] proved an affirmative analog to Bukh's question when each instance of χ is replaced by the fractional chromatic number, χ f . Though incomparable, these results are related by the general fact that |V (H)|/α(H) ≤ χ f (H) ≤ χ(H) for all H. Thus, these affirmatively resolve question 1 for any graph for which n/α(G) (or somewhat more generally χ f (G)) is within a multiplicative factor of χ(G), which by [5] includes almost all graphs.…”

Combinatorial and geometric dualities in graph homomorphism optimization problems

“…First, the fractional chromatic number and the Lovász theta function provide the clearest connection between the combinatorial framework presented in this chapter and the geometric framework presented in Chapter 4. Second, there is significant interest from the graph theory community on χ f , as there are conjectures which are either open or false when stated in terms of χ, but are known to be true when stated in terms of χ f (see [89,90,110,122]). We make no attempt at a comprehensive survey on the fractional chromatic number: we have simply selected results relevant to the themes of this text.…”