For many pairs of degrees, the random lower-degree regular graph can be embedded inside the random higher-degree regular graph with high probability, and unions of random regular graphs can mimic a single random regular graph.
Gao, The number of perfect matchings, and the nesting properties, of random regular graphs, Random Structures and Algorithms62 (2023), no
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Monotonicity and decompositions of random regular graphs
For many pairs of degrees, the random lower-degree regular graph can be embedded inside the random higher-degree regular graph with high probability, and unions of random regular graphs can mimic a single random regular graph.