Define the *divisibility graph* of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other.


For all *N*, is it true that integers less than or equal to *N* whose proper divisors have divisibility graph which is planar are more numerous than those that don't?


Using SAGE, Freddy Barrera determined those not greater than 1000 which are not planar:


32, 36, 48, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 234, 240, 243, 252, 256, 260, 264, 270, 272, 276, 280, 288, 294, 300, 304, 306, 308, 312, 315, 320, 324, 330, 336, 340, 342, 348, 350, 352, 360, 364, 368, 372, 378, 380, 384, 390, 392, 396, 400, 405, 408, 414, 416, 420, 432, 440, 441, 444, 448, 450, 456, 460, 462, 464, 468, 476, 480, 484, 486, 490, 492, 495, 496, 500, 504, 510, 512, 516, 520, 522, 525, 528, 532, 540, 544, 546, 550, 552, 558, 560, 564, 567, 570, 572, 576, 580, 585, 588, 592, 594, 600, 608, 612, 616, 620, 624, 630, 636, 640, 644, 648, 650, 656, 660, 666, 672, 675, 676, 680, 684, 688, 690, 693, 696, 700, 702, 704, 708, 714, 720, 726, 728, 729, 732, 735, 736, 738, 740, 744, 748, 750, 752, 756, 760, 765, 768, 770, 774, 780, 784, 792, 798, 800, 804, 810, 812, 816, 819, 820, 825, 828, 832, 836, 840, 846, 848, 850, 852, 855, 858, 860, 864, 868, 870, 876, 880, 882, 884, 888, 891, 896, 900, 910, 912, 918, 920, 924, 928, 930, 936, 940, 944, 945, 948, 950, 952, 954, 960, 966, 968, 972, 975, 976, 980, 984, 988, 990, 992, 996, 1000


<https://puzzling.stackexchange.com/questions/112686/the-divisibility-graph-again/112731#112731>



Is the divisibility graph of the proper divisors of n more often planar than not?
No, because almost all numbers have at least $4$ distinct prime factors, making the divisibility graph contain a hypercube and thus be nonplanar.


