The t-Improper Chromatic Number of Random Graphs

Abstract

We consider the t-improper chromatic number of the Erdős-Rényi random graph G n,p . The timproper chromatic number χ t (G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we provide a detailed description of the asymptotic behaviour of χ t (G n,p ) over the range of choices for the growth of t = t(n).

Largest sparse subgraphs of random graphs

“…(2.4) in [10], while the lower bound is implied by a sharp form of Stirling's formula, e.g. (1.4) of [3]: see the appendix of [13] for an explicit proof (when r is integral).…”

“…[12, Lemma 2.2]. If t = Θ(ln(np)), then the growth of the first-order term ofα t (G n,p ) is a multiple of log b (np), and large deviation techniques were used to determine the factor (which depends on p and t) [13]. (With the exception of the precise factor at the threshold t = Θ(ln(np)), these statements have been shown to remain valid for smaller values of p as long as p ≫ 1/n, cf.…”

“…also [2]), for partitions of random graphs according to a fixed hereditary property, are not applicable here. In our previous studies [8,13], it was useful that t-dependence is hereditary for fixed t. Unfortunately, this is not the case for t-sparsity.…”

Coloring random graphs online without creating monochromatic subgraphs

“…Very precise bounds on the chromatic number of the random graph are known by now (see e.g. ), and several related coloring notions and their associated ‘chromatic numbers’ have been investigated for the random graph .…”

“…Note that a proper coloring in the usual sense is a coloring that is valid with respect to a single edge. More generally, a coloring that is valid with respect to the star with rays is a coloring in which each color class induces a graph with maximum degree at most (this is sometimes called an ‐improper coloring, see and references therein).…”