On Graphical Partitions with Restricted Parts

An integer partition of $n$ is called graphical if its parts form a degree sequence of a simple graph. While unrestricted graphical partitions have been extensively studied, much less is known when the parts are restricted to a prescribed set. In this work, we investigate the probability that a uniformly random partition of an even integer $n$, subject to such restrictions, is graphical. We establish an upper bound on this probability expressed solely in terms of the Durfee square of the partition. Additionally, letting $p_g(n)$ denote the probability that a random restricted partition of an even integer $n$ is graphical, we prove that the limit inferior of $p_g(n)$ is 0. Furthermore, we obtain an explicit bound on the decay rate of $p_g(n)$ in terms of $n$ and the imposed restrictions on the parts. Our approach employs the Nash-Williams graphical condition, the saddle-point method and Edgeworth expansions.