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
Best Answer
No, because almost all numbers have at least $4$ distinct prime factors, making the divisibility graph contain a hypercube and thus be nonplanar.