We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree ∆ are list (∆ + 1)-...

We provide a large deviation result for a random sum ∑Nx n=0Xn, where Nx is a renewal counting process and Xn n≥0 are i.i.d. random variables, independent of Nx , with a common distribution that belongs to a class of square root insensitive distributions. Asymptotically, the tails of these distributions are heavier than e− √ x and have zero relative decrease in intervals of length √ x, hence sq...