======================================================================== Subgraph Lattice Theorem — Verification Checking: E(G) ⊆ E(H) âŸč p_đ”Ÿ(G) ≀ p_đ”Ÿ(H) ======================================================================== n = 4: 8 graphs Containment chains at n=4: P_n (path)(p= 8, m= 3) ⊂ C_n (cycle)(p= 12, m= 4) ✓ P_n (path)(p= 8, m= 3) ⊂ W_n (wheel)(p= 15, m= 6) ✓ P_n (path)(p= 8, m= 3) ⊂ Fan_n(p= 13, m= 5) ✓ P_n (path)(p= 8, m= 3) ⊂ K_n (complete)(p= 15, m= 6) ✓ S_n (star)(p= 8, m= 3) ⊂ W_n (wheel)(p= 15, m= 6) ✓ S_n (star)(p= 8, m= 3) ⊂ K_n (complete)(p= 15, m= 6) ✓ C_n (cycle)(p= 12, m= 4) ⊂ W_n (wheel)(p= 15, m= 6) ✓ C_n (cycle)(p= 12, m= 4) ⊂ Fan_n(p= 13, m= 5) ✓ C_n (cycle)(p= 12, m= 4) ⊂ K_n (complete)(p= 15, m= 6) ✓ K_{2,n-2}(p= 12, m= 4) ⊂ W_n (wheel)(p= 15, m= 6) ✓ Fan_n(p= 13, m= 5) ⊂ W_n (wheel)(p= 15, m= 6) ✓ Ladder_n(p= 12, m= 4) ⊂ W_n (wheel)(p= 15, m= 6) ✓ K_{2,n-2}(p= 12, m= 4) ⊂ K_n (complete)(p= 15, m= 6) ✓ Fan_n(p= 13, m= 5) ⊂ K_n (complete)(p= 15, m= 6) ✓ Ladder_n(p= 12, m= 4) ⊂ K_n (complete)(p= 15, m= 6) ✓ n = 5: 8 graphs Containment chains at n=5: P_n (path)(p= 16, m= 4) ⊂ C_n (cycle)(p= 27, m= 5) ✓ P_n (path)(p= 16, m= 4) ⊂ W_n (wheel)(p= 43, m= 8) ✓ P_n (path)(p= 16, m= 4) ⊂ Fan_n(p= 34, m= 7) ✓ P_n (path)(p= 16, m= 4) ⊂ K_n (complete)(p= 52, m=10) ✓ S_n (star)(p= 16, m= 4) ⊂ W_n (wheel)(p= 43, m= 8) ✓ S_n (star)(p= 16, m= 4) ⊂ Friendship_n(p= 25, m= 6) ✓ S_n (star)(p= 16, m= 4) ⊂ K_n (complete)(p= 52, m=10) ✓ C_n (cycle)(p= 27, m= 5) ⊂ W_n (wheel)(p= 43, m= 8) ✓ C_n (cycle)(p= 27, m= 5) ⊂ Fan_n(p= 34, m= 7) ✓ C_n (cycle)(p= 27, m= 5) ⊂ K_n (complete)(p= 52, m=10) ✓ Friendship_n(p= 25, m= 6) ⊂ W_n (wheel)(p= 43, m= 8) ✓ W_n (wheel)(p= 43, m= 8) ⊂ K_n (complete)(p= 52, m=10) ✓ K_{2,n-2}(p= 34, m= 6) ⊂ K_n (complete)(p= 52, m=10) ✓ Fan_n(p= 34, m= 7) ⊂ K_n (complete)(p= 52, m=10) ✓ Friendship_n(p= 25, m= 6) ⊂ K_n (complete)(p= 52, m=10) ✓ n = 6: 9 graphs Containment chains at n=6: P_n (path)(p= 32, m= 5) ⊂ C_n (cycle)(p= 58, m= 6) ✓ P_n (path)(p= 32, m= 5) ⊂ W_n (wheel)(p= 118, m=10) ✓ P_n (path)(p= 32, m= 5) ⊂ Fan_n(p= 89, m= 9) ✓ P_n (path)(p= 32, m= 5) ⊂ K_n (complete)(p= 203, m=15) ✓ S_n (star)(p= 32, m= 5) ⊂ W_n (wheel)(p= 118, m=10) ✓ S_n (star)(p= 32, m= 5) ⊂ K_n (complete)(p= 203, m=15) ✓ C_n (cycle)(p= 58, m= 6) ⊂ W_n (wheel)(p= 118, m=10) ✓ C_n (cycle)(p= 58, m= 6) ⊂ Fan_n(p= 89, m= 9) ✓ C_n (cycle)(p= 58, m= 6) ⊂ K_n (complete)(p= 203, m=15) ✓ W_n (wheel)(p= 118, m=10) ⊂ K_n (complete)(p= 203, m=15) ✓ K_{2,n-2}(p= 96, m= 8) ⊂ K_n (complete)(p= 203, m=15) ✓ Fan_n(p= 89, m= 9) ⊂ K_n (complete)(p= 203, m=15) ✓ Ladder_n(p= 74, m= 7) ⊂ Prism_n(p= 114, m= 9) ✓ Ladder_n(p= 74, m= 7) ⊂ K_n (complete)(p= 203, m=15) ✓ Prism_n(p= 114, m= 9) ⊂ K_n (complete)(p= 203, m=15) ✓ n = 7: 8 graphs Containment chains at n=7: P_n (path)(p= 64, m= 6) ⊂ C_n (cycle)(p= 121, m= 7) ✓ P_n (path)(p= 64, m= 6) ⊂ W_n (wheel)(p= 316, m=12) ✓ P_n (path)(p= 64, m= 6) ⊂ Fan_n(p= 233, m=11) ✓ P_n (path)(p= 64, m= 6) ⊂ K_n (complete)(p= 877, m=21) ✓ S_n (star)(p= 64, m= 6) ⊂ W_n (wheel)(p= 316, m=12) ✓ S_n (star)(p= 64, m= 6) ⊂ Friendship_n(p= 125, m= 9) ✓ S_n (star)(p= 64, m= 6) ⊂ K_n (complete)(p= 877, m=21) ✓ C_n (cycle)(p= 121, m= 7) ⊂ W_n (wheel)(p= 316, m=12) ✓ C_n (cycle)(p= 121, m= 7) ⊂ Fan_n(p= 233, m=11) ✓ C_n (cycle)(p= 121, m= 7) ⊂ K_n (complete)(p= 877, m=21) ✓ Friendship_n(p= 125, m= 9) ⊂ W_n (wheel)(p= 316, m=12) ✓ W_n (wheel)(p= 316, m=12) ⊂ K_n (complete)(p= 877, m=21) ✓ K_{2,n-2}(p= 274, m=10) ⊂ K_n (complete)(p= 877, m=21) ✓ Fan_n(p= 233, m=11) ⊂ K_n (complete)(p= 877, m=21) ✓ Friendship_n(p= 125, m= 9) ⊂ K_n (complete)(p= 877, m=21) ✓ n = 8: 9 graphs Containment chains at n=8: P_n (path)(p= 128, m= 7) ⊂ C_n (cycle)(p= 248, m= 8) ✓ P_n (path)(p= 128, m= 7) ⊂ W_n (wheel)(p= 836, m=14) ✓ P_n (path)(p= 128, m= 7) ⊂ Fan_n(p= 610, m=13) ✓ P_n (path)(p= 128, m= 7) ⊂ K_n (complete)(p= 4140, m=28) ✓ S_n (star)(p= 128, m= 7) ⊂ W_n (wheel)(p= 836, m=14) ✓ S_n (star)(p= 128, m= 7) ⊂ K_n (complete)(p= 4140, m=28) ✓ C_n (cycle)(p= 248, m= 8) ⊂ W_n (wheel)(p= 836, m=14) ✓ C_n (cycle)(p= 248, m= 8) ⊂ Fan_n(p= 610, m=13) ✓ C_n (cycle)(p= 248, m= 8) ⊂ K_n (complete)(p= 4140, m=28) ✓ W_n (wheel)(p= 836, m=14) ⊂ K_n (complete)(p= 4140, m=28) ✓ K_{2,n-2}(p= 792, m=12) ⊂ K_n (complete)(p= 4140, m=28) ✓ Fan_n(p= 610, m=13) ⊂ K_n (complete)(p= 4140, m=28) ✓ Ladder_n(p= 456, m=10) ⊂ Prism_n(p= 958, m=12) ✓ Ladder_n(p= 456, m=10) ⊂ K_n (complete)(p= 4140, m=28) ✓ Prism_n(p= 958, m=12) ⊂ K_n (complete)(p= 4140, m=28) ✓ n = 9: 8 graphs Containment chains at n=9: P_n (path)(p= 256, m= 8) ⊂ C_n (cycle)(p= 503, m= 9) ✓ P_n (path)(p= 256, m= 8) ⊂ W_n (wheel)(p= 2199, m=16) ✓ P_n (path)(p= 256, m= 8) ⊂ Fan_n(p= 1597, m=15) ✓ P_n (path)(p= 256, m= 8) ⊂ K_n (complete)(p= 21147, m=36) ✓ S_n (star)(p= 256, m= 8) ⊂ W_n (wheel)(p= 2199, m=16) ✓ S_n (star)(p= 256, m= 8) ⊂ Friendship_n(p= 625, m=12) ✓ S_n (star)(p= 256, m= 8) ⊂ K_n (complete)(p= 21147, m=36) ✓ C_n (cycle)(p= 503, m= 9) ⊂ W_n (wheel)(p= 2199, m=16) ✓ C_n (cycle)(p= 503, m= 9) ⊂ Fan_n(p= 1597, m=15) ✓ C_n (cycle)(p= 503, m= 9) ⊂ K_n (complete)(p= 21147, m=36) ✓ Friendship_n(p= 625, m=12) ⊂ W_n (wheel)(p= 2199, m=16) ✓ W_n (wheel)(p= 2199, m=16) ⊂ K_n (complete)(p= 21147, m=36) ✓ K_{2,n-2}(p= 2314, m=14) ⊂ K_n (complete)(p= 21147, m=36) ✓ Fan_n(p= 1597, m=15) ⊂ K_n (complete)(p= 21147, m=36) ✓ Friendship_n(p= 625, m=12) ⊂ K_n (complete)(p= 21147, m=36) ✓ n = 10: 10 graphs Containment chains at n=10: P_n (path)(p= 512, m= 9) ⊂ C_n (cycle)(p= 1014, m=10) ✓ P_n (path)(p= 512, m= 9) ⊂ W_n (wheel)(p= 5769, m=18) ✓ P_n (path)(p= 512, m= 9) ⊂ Fan_n(p= 4181, m=17) ✓ P_n (path)(p= 512, m= 9) ⊂ K_n (complete)(p=115975, m=45) ✓ S_n (star)(p= 512, m= 9) ⊂ W_n (wheel)(p= 5769, m=18) ✓ S_n (star)(p= 512, m= 9) ⊂ K_n (complete)(p=115975, m=45) ✓ C_n (cycle)(p= 1014, m=10) ⊂ W_n (wheel)(p= 5769, m=18) ✓ C_n (cycle)(p= 1014, m=10) ⊂ Fan_n(p= 4181, m=17) ✓ C_n (cycle)(p= 1014, m=10) ⊂ K_n (complete)(p=115975, m=45) ✓ W_n (wheel)(p= 5769, m=18) ⊂ K_n (complete)(p=115975, m=45) ✓ K_{2,n-2}(p= 6816, m=16) ⊂ K_n (complete)(p=115975, m=45) ✓ Fan_n(p= 4181, m=17) ⊂ K_n (complete)(p=115975, m=45) ✓ Ladder_n(p= 2810, m=13) ⊂ Prism_n(p= 7076, m=15) ✓ Ladder_n(p= 2810, m=13) ⊂ K_n (complete)(p=115975, m=45) ✓ Prism_n(p= 7076, m=15) ⊂ K_n (complete)(p=115975, m=45) ✓ Petersen(p= 8581, m=15) ⊂ K_n (complete)(p=115975, m=45) ✓ ======================================================================== RESULTS ======================================================================== Total ordered pairs checked: 458 Comparable (E(G) ⊆ E(H)): 108 Incomparable pairs: 244 VIOLATIONS: 0 ✓ THEOREM CONFIRMED on all 108 comparable pairs. (Plus 244 incomparable pairs where the theorem makes no prediction.) ======================================================================== Îș-COUNTEREXAMPLES — confirming incomparability ======================================================================== n=7: Friendship_n vs C_n (cycle): INCOMPARABLE ✓ Edges only in Friendship_n: 4 Edges only in C_n (cycle): 2 n=9: Friendship_n vs C_n (cycle): INCOMPARABLE ✓ Edges only in Friendship_n: 6 Edges only in C_n (cycle): 3 n=9: K_{2,n-2} vs W_n (wheel): INCOMPARABLE ✓ Edges only in K_{2,n-2}: 5 Edges only in W_n (wheel): 7 n=10: K_{2,n-2} vs W_n (wheel): INCOMPARABLE ✓ Edges only in K_{2,n-2}: 6 Edges only in W_n (wheel): 8