Higher Gonalities of Erdős-Rényi Random Graphs

We consider the asymptotic behavior of the second and higher gonalities of an Erdős-Rényi random graph and provide upper bounds for both via the probabilistic method. Our results suggest that for sufficiently large n, the second gonality of an Erdős-Rényi random Graph G(n, p) is strictly less than and asymptotically equal to the number of vertices under a suitable restriction of the probability p. We also prove an asymptotic upper bound for all higher gonalities of large Erdős-Rényi random graphs that adapts and generalizes a similar result on complete graphs. We suggest another approach towards finding both upper and lower bounds for the second and higher gonalities for small p = c n , using a special case of the Riemann-Roch Theorem, and fully determine the asymptotic behavior of arbitrary gonalities when c ≤ 1.