Davies, Ewan, Jenssen, Matthew, Perkins, Will and Roberts, Barnaby
(2017)
On the average size of independent sets in trianglefree graphs.
Proceedings of the American Mathematical Society, 146 (1).
pp. 111124.
ISSN 00029939
Abstract
We prove an asymptotically tight lower bound on the average size of independent sets in a trianglefree graph on $ n$ vertices with maximum degree $ d$, showing that an independent set drawn uniformly at random from such a graph has expected size at least $ (1+o_d(1)) \frac {\log d}{d}n$. This gives an alternative proof of Shearer's upper bound on the Ramsey number $ R(3,k)$. We then prove that the total number of independent sets in a trianglefree graph with maximum degree $ d$ is at least $ \exp \left [\left (\frac {1}{2}+o_d(1) \right ) \frac {\log ^2 d}{d}n \right ]$. The constant $ 1/2$ in the exponent is best possible. In both cases, tightness is exhibited by a random $ d$regular graph. Both results come from considering the hardcore model from statistical physics: a random independent set $ I$ drawn from a graph with probability proportional to $ \lambda ^{\vert I\vert}$, for a fugacity parameter $ \lambda >0$. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hardcore model on trianglefree graphs of maximum degree $ d$. The bound is asymptotically tight in $ d$ for all $ \lambda =O_d(1)$. We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a trianglefree graph and give some consequences of these conjectures in Ramsey theory
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