Prescribed Degree Distributions in Chordal Graphs of Bounded Clique Number

For every fixed integer k ≥ 1 we determine which probability laws p on {0, 1, 2, …} arise as limiting degree distributions of chordal graphs with clique number at most k + 1: exactly the laws with E_p[D] ≤ 2 E_p[min(D, k)]. The same condition characterizes k-degenerate graphs and graphs of treewidth or pathwidth at most k, both for pointwise limits and for strong limits in which the mean degree converges as well. We then map where this collapse stops. We characterize the limit laws of k-trees, of connected graphs and of proper interval graphs, and we show that interval graphs break away for every k ≥ 2: on the boundary of the feasible region a law is an interval limit law if and only if it has no mass on {1, …, k − 1}, and this exclusion is robust in Wasserstein distance. The proof rests on a clique-tree deficit inequality for chordal graphs. Sufficiency is constructive. Vertices arrive on a line and each one "throws" a ball a few steps ahead, as in juggling notation; three degree-balancing blocks keep the active window at exactly k vertices. The output carries a perfect elimination order and a subtree representation and is built in O(kn) time. With a deterministic quota choice of blocks, every output on n vertices satisfies an explicit deterministic bound on its degree-count error, and twice its number of edges is within 30k³ of n·E_p[D], whatever the tail. For pure power laws p_d ∝ d^(−γ) the total-variation error is Θ(n^(−(γ−1)/γ)) for fixed k, which no graph on n vertices can beat, and no degree cut-off is needed. We also determine the smallest feasible power-law exponent for each k, and show that for k = 1 pathwidth one forces linear diameter while trees with the same degree multiset have logarithmic diameter.