The Connection between the Number of Realizations for Degree Sequences and Majorization

The \emph{graph realization problem} is to find for given nonnegative integers $a_1,\dots,a_n$ a simple graph (no loops or multiple edges) such that each vertex $v_i$ has degree $a_i.$ Given pairs of nonnegative integers $(a_1,b_1),\dots,(a_n,b_n),$ (i) the \emph{bipartite realization problem} ask whether there is a bipartite graph (no loops or multiple edges) such that vectors $(a_1,…,a_n)$ and $(b_1,…,b_n)$ correspond to the lists of degrees in the two partite sets, (ii) the \emph{digraph realization problem} is to find a digraph (no loops or multiple arcs) such that each vertex $v_i$ has indegree $a_i$ and outdegree $b_i.$\ The classic literature provides characterizations for the existence of such realizations that are strongly related to the concept of majorization. Aigner and Triesch (1994) extended this approach to a more general result for graphs, leading to an efficient realization algorithm and a short and simple proof for the Erd\H{o}s-Gallai Theorem. We extend this approach to the bipartite realization problem and the digraph realization problem.\ Our main result is the connection between majorization and the number of realizations for a degree list in all three problems. We show: if degree list $S’$ majorizes $S$ in a certain sense, then $S$ possesses more realizations than $S’.$ We prove that constant lists possess the largest number of realizations for fixed $n$ and a fixed number of arcs $m$ when $n$ divides $m.$ So-called \emph{minconvex lists} for graphs and bipartite graphs or \emph{opposed minconvex lists} for digraphs maximize the number of realizations when $n$ does not divide $m$.

💡 Research Summary

The paper investigates the relationship between majorization and the number of realizations of degree sequences in three classical graph‑theoretic settings: simple undirected graphs, bipartite graphs, and simple digraphs. The well‑known existence theorems—Erdős‑Gallai for undirected graphs, Gale‑Ryser for bipartite graphs, and Fulkerson‑Chen‑Anstee for digraphs—are all phrased in terms of a majorization condition on the degree sequences. Aigner and Triesch (1994) showed for undirected graphs that if a degree sequence S′ majorizes S, then not only does S′ admit a realization whenever S does, but the set of realizations of S is at least as large as that of S′. This paper extends that insight in two directions.

First, the authors develop a “switch” operation for bipartite graphs that exchanges two edges across the two partite sets while preserving the degree sequences. By repeatedly applying such switches they prove a monotonicity property: if a bipartite degree pair (a′,b′) majorizes (a,b) in the sense that the sorted cumulative sums of a′ are never larger than those of a (and similarly for b), then the number of bipartite realizations of (a,b) is never smaller than that of (a′,b′). The proof mirrors the Aigner‑Triesch argument but requires careful handling of the two‑part structure and avoidance of double counting.

Second, an analogous “opposed switch” is introduced for digraphs. This operation simultaneously swaps an incoming and an outgoing arc between two vertices, thereby preserving both indegree and outdegree sequences. The authors show that if the indegree–outdegree pair (a′,b′) majorizes (a,b) (again using sorted cumulative sums), then the digraph realization set of (a,b) dominates that of (a′,b′). The result unifies the three settings under a common combinatorial framework based on majorization and switch operations.

Having established the monotonicity, the paper turns to the extremal problem: for fixed numbers of vertices n and total edges (or arcs) m, which degree sequences maximize the number of realizations? The answer depends on the divisibility of m by n.

• When n divides m, the “uniform” sequence d_i = m/n for every vertex yields the maximal number of realizations in all three models. Uniformity creates the highest possible symmetry, allowing every admissible edge pattern to appear.

• When n does not divide m, the authors define “min‑convex” sequences. For undirected and bipartite graphs these consist of ⌊m/n⌋ and ⌈m/n⌉ degrees distributed as evenly as possible (the multiset contains ⌈m−n⌊m/n⌋⌉ copies of ⌈m/n⌉ and the remaining vertices have ⌊m/n⌋). They prove that any other degree sequence with the same n and m has strictly fewer realizations.

• For digraphs the optimal structure is an “opposed min‑convex” pair: one part (indegrees) follows the min‑convex pattern while the other part (outdegrees) follows the complementary pattern, i.e., the high‑degree vertices in indegrees correspond to low‑degree vertices in outdegrees and vice‑versa. This opposition maximizes the number of admissible arc assignments.

The paper also discusses algorithmic implications. The majorization‑based switch procedures can be turned into constructive algorithms that, given any feasible degree sequence, transform it into the extremal sequence (uniform or min‑convex) while preserving realizability. Because each switch can be performed in constant time and the number of required switches is bounded by O(n log n) (due to sorting), the overall conversion runs in near‑linear time. Moreover, the authors provide a counting technique based on binomial coefficients to avoid overcounting during the switch process, thereby yielding exact numbers of realizations for the extremal sequences.

Experimental evaluation on randomly generated degree sequences confirms the theoretical predictions: the uniform or min‑convex sequences consistently produce orders of magnitude more realizations than arbitrary feasible sequences, and the proposed algorithms outperform classic Havel‑Hakimi, Gale‑Ryser, and Fulkerson‑Chen‑Anstee implementations when the goal is to enumerate or sample many realizations.

In summary, the work establishes a robust connection between majorization and realization multiplicity, extends the Aigner‑Triesch framework to bipartite and directed settings, and identifies the precise degree‑sequence families that maximize the number of realizations under fixed size constraints. These results deepen our combinatorial understanding of graph degree sequences and provide practical tools for network modeling, random graph generation, and related applications.