Degree sequences realizing labelled perfect matchings

Abstract

Let $n\in \mathbb{N}$ and $d_1 \geq d_2 \geq d_n\geq 1$ be integers. There is characterization of when $(d_1, d_1, \ldots, d_n)$ is the degree sequence of a graph containing a perfect matching, due to results of Lov\'{a}sz (1974) and Erd\H{o}s and Gallai (1960). But \emph{which} perfect matchings can be realized in the labelled graph? Here we find the extremal answers to this question, showing that the sequence $(d_1, d_2, \ldots, d_n)$: (1) can realize a perfect matching iff it can realize $\{(1, n), (2,n-1), \ldots, (n/2, n/2+1)\}$, and; (2) can realize any perfect matching iff it can realize $\{(1, 2), (3,4), \ldots, (n-1, n)\}$. Our main result is a characterization of when (2) occurs, extending the work of Lov\'{a}sz and Erd\H{o}s and Gallai. Separately, we are also able to establish a conjecture of Yin and Busch, Ferrera, Hartke, Jacobsen, Kaul, and West about packing graphic sequences, establishing a degree-sequence analog of the Sauer-Spencer packing theorem. We conjecture an $h$-factor analog of our main result, and discuss implications for packing $h$ disjoint perfect matchings.