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arXiv · 0812.2937

On the chromatic number of random d-regular graphs

Graeme Kemkes·Xavier Pérez-Giménez·Nicholas Wormald
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Abstract

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal.

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Graeme Kemkes, Xavier Pérez-Giménez, Nicholas Wormald. 2008-12-15. On the chromatic number of random d-regular graphs. https://arxiv.org/abs/0812.2937

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