Lego-like spheres and tori

Given a connected surface F 2 with Euler characteristic χ and three integers b > a ≥ 1 < k , an ( { a , b } ; k ) - F 2 is a F 2 -embedded graph, having vertices of degree only k and only a - and b -gonal faces. The main case are (geometric) fullerenes (5, 6; 3)- S 2 . By p a , p b we denote the number of a -gonal, b -gonal faces. Call an ( { a , b } ; k ) -map lego-admissible if either p b p a , or p a p b is integer. Call it lego-like if it is either a b f - lego map , or a f b - lego map , i.e., the face-set is partitioned into min ( p a , p b ) isomorphic clusters, legos , consisting either one a -gon and f = p b p a b -gons, or, respectively, f = p a p b a -gons and one b -gon; the case f = 1 we denote also by ab . Call a ( { a , b } ; k ) -map elliptic , parabolic or hyperbolic if the curvature κ b = 1 + b k - b 2 of b -gons is positive, zero or negative, respectively. There are 14 lego-like elliptic ( { a , b } ; k ) - S 2 with ( a , b ) ≠ ( 1 , 2 ) . No ( { 1 , 3 } ; 6 ) - S 2 is lego-admissible. For other 7 families of parabolic ( { a , b } ; k ) - S 2 , each lego-admissible sphere with p a ≤ p b is a f b and an infinity (by Goldberg–Coxeter operation ) of a b f -spheres exist. The number of hyperbolic a b f ( { a , b } ; k ) - S 2 with ( a , b ) ≠ ( 1 , 3 ) is finite. Such a f b -spheres with a ≥ 3 have ( a , k ) = ( 3 , 4 ) , ( 3 , 5 ) , ( 4 , 3 ) , ( 5 , 3 ) or (3, 3); their number is finite for each b , but infinite for each of 5 cases ( a , k ). Any lego-admissible ( { a , b } ; k ) - S 2 with p b = 2 ≤ a is a f b . We list, explicitly or by parameters, lego-admissible ( { a , b } ; k ) -maps among: hyperbolic spheres, spheres with a ∈ { 1 , 2 } , spheres with p b ∈ { 2 , p a 2 } , Goldberg–Coxeter’s spheres and ( { a , b } ; k ) -tori. We present extensive computer search of lego-like spheres: 7 parabolic ( p b -dependent) families, basic examples of all 5 hyperbolic a f b ( b -dependent) families with a ≥ 3 , and lego-like ( { a , b } ; 3 ) -tori.