Abstract
We prove an asymptotic formula for the number of orientations with given out-degree (score) sequence for a graph $G$. The graph $G$ is assumed to have average degrees at least $n^{1/3 + \epsilon}$ for some $\epsilon > 0$, and to have strong mixing properties, while the maximum imbalance (out-degree minus in-degree) of the orientation should be not too large. Our enumeration results have applications to the study of subdigraph occurrences in random orientations with given imbalance sequence. As one step of our calculation, we obtain new bounds for the maximum likelihood estimators for the Bradley-Terry model of paired comparisons.
Highlights
Let G be an undirected simple graph with vertices {1, 2, . . . , n}
Our primary aim in this paper is to find the asymptotic number of orientations of G
In order to apply the saddle point method to enumerate the number of orientations, we will use the standard parameters in the Bradley-Terry model of paired comparisons
Summary
Let G be an undirected simple graph with vertices {1, 2, . . . , n}. An orientation of G is an assignment of one of the two possible directions to each edge, thereby making an oriented graph G. Our primary aim in this paper is to find the asymptotic number of orientations of G with given imbalance sequence. In solving this enumeration problem, we will apply the saddle point method to a suitable generating function, using Cauchy’s Theorem while following the general framework outlined in [12]. Theorem 1 and Lemma 2 immediately give an asymptotic formula for the number of Eulerian orientations. Applications of these results include estimating the probability for a uniform random orientation with given imbalance sequence to contain a prescribed subdigraph. The proof of Theorem 4 is given at the end of Section 2
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