Colouring random graphs: Tame colourings
Annika Heckel, Konstantinos Panagiotou

TL;DR
This paper investigates the concentration properties of t-bounded colourings in random graphs, revealing two-point concentration results and their implications for the distribution of the chromatic number.
Contribution
It introduces the concept of tame colourings and provides tight probabilistic bounds, advancing understanding of colourings in random graphs and supporting the Zigzag Conjecture.
Findings
hi_t(G(n,m)) is maximally concentrated on at most two explicit values for t=lpha(G(n,m))-2
hi_t(G_{n,1/2}) is contained in a length n^{0.99} interval with high probability under certain conditions
Proves two-point concentration of the equitable chromatic number of G(n,m)
Abstract
Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded chromatic number of G, denoted by , is the minimum number of colours in such a colouring. Every colouring of G is then -bounded, where denotes the size of a largest independent set. We study colourings of the random graph G(n, 1/2) and of the corresponding uniform random graph G(n,m) with . We show that is maximally concentrated on at most two explicit values for . This behaviour stands in stark contrast to that of the normal chromatic number, which was recently shown not to be concentrated on any sequence of intervals of length…
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Taxonomy
TopicsLimits and Structures in Graph Theory
