article published yes Lawrence Hollom author Lyuben Lichev author 9aa8388e-d003-11ee-8458-c4c1d7447977 Adva Mond author Julien Portier author Yiting Wang author 1917d194-076e-11ed-97cd-837255f88785 MaKw department GradSch department Randomness and structure in combinatorics project THRESHOLDS IN SEMI-RANDOM PROCESSES AND GRAPH EMBEDDINGS project In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters d1 ≤ d2, there exists a coupling of G(n,d1) and G(n,d2) satisfying that G(n,d1) ⊆ G(n,d2) with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime. Our contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for generating unions of random edge-disjoint perfect matchings, and refined estimates of Gao’s bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev, McKay, Southwell and Zhukovskii. https://research-explorer.ista.ac.at/download/23155/23165/2026_AnnalsAppliedProbability_Hollom.pdf application/pdfno Institute of Mathematical Statistics2026 eng Random regular graphscouplingcontiguitytotal variation distancedecomposition 1050-5164 2168-8737 2505.2287510.1214/26-AAP2345 3654716-4738 Hollom L, Lichev L, Mond A, Portier J, Wang Y. 2026. Monotonicity and decompositions of random regular graphs. Annals of Applied Probability. 36(5), 4716–4738. Hollom, L., Lichev, L., Mond, A., Portier, J., &#38; Wang, Y. (2026). Monotonicity and decompositions of random regular graphs. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/26-AAP2345">https://doi.org/10.1214/26-AAP2345</a> Hollom L, Lichev L, Mond A, Portier J, Wang Y. Monotonicity and decompositions of random regular graphs. <i>Annals of Applied Probability</i>. 2026;36(5):4716-4738. doi:<a href="https://doi.org/10.1214/26-AAP2345">10.1214/26-AAP2345</a> L. Hollom, L. Lichev, A. Mond, J. Portier, and Y. Wang, “Monotonicity and decompositions of random regular graphs,” <i>Annals of Applied Probability</i>, vol. 36, no. 5. Institute of Mathematical Statistics, pp. 4716–4738, 2026. Hollom, Lawrence, Lyuben Lichev, Adva Mond, Julien Portier, and Yiting Wang. “Monotonicity and Decompositions of Random Regular Graphs.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2026. <a href="https://doi.org/10.1214/26-AAP2345">https://doi.org/10.1214/26-AAP2345</a>. L. Hollom, L. Lichev, A. Mond, J. Portier, Y. Wang, Annals of Applied Probability 36 (2026) 4716–4738. Hollom, Lawrence, et al. “Monotonicity and Decompositions of Random Regular Graphs.” <i>Annals of Applied Probability</i>, vol. 36, no. 5, Institute of Mathematical Statistics, 2026, pp. 4716–38, doi:<a href="https://doi.org/10.1214/26-AAP2345">10.1214/26-AAP2345</a>. 231552026-10-18T22:01:52Z2026-10-19T07:39:17Z