A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S i...A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating.Let k be a positive integer,S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern.Moreover,if graph S is k-resonant for any integer k,then S is called maximally resonant.In this paper,we show that the maximally resonant{(3,4),4}-fullerene graphs are S_6,S_8,S_(10)^(2),S_(12)^(2),S_(12)^(4),S_(12)^(5),S_(14)^(3),S_(14)^(5),S_(16)^(3),S_(18)^(5),S_(24)as shown in Fig.1.As a corollary,it is shown that if a{(3,4),4}-fullerene graph is 4-resonant,then it is also maximally resonant.展开更多
A(3,6)-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons,and has the connectivity 2 or 3.The(3,6)-fullerenes with connectivity 2 are the tubes consisting of l concentric hexagonal ...A(3,6)-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons,and has the connectivity 2 or 3.The(3,6)-fullerenes with connectivity 2 are the tubes consisting of l concentric hexagonal layers such that each layer consists of two hexangons,capped on each end by two adjacent triangles,denoted by T_(l)(l≥1).A(3,6)-fullerene Tl with n vertices has exactly 2n/4+1 perfect matchings.The structure of a(3,6)-fullerene G with connectivity 3 can be determined by only three parameters r,s and t,thus we denote it by G=(r,s,t),where r is the radius(number of rings),s is the size(number of spokes in each layer,s(≥4,s is even),and t is the torsion(0≤t<s,t≡r mod 2).In this paper,the counting formula of the perfect matchings in G=n+1,4,t)is given,and the number of perfect matchpings is obtained.Therefore,the correctness of the conclusion that every bridgeless cubic graph with p vertices has at least 2p/3656perfect matchings proposed by Esperet et al is verified for(3,6)-fullerene G=(n+1,4,t).展开更多
基金Supported by NSFC(Grant Nos.11801148 and 11626089)the Foundation for the Doctor of Henan Polytechnic University(Grant No.B2014-060)。
文摘A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating.Let k be a positive integer,S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern.Moreover,if graph S is k-resonant for any integer k,then S is called maximally resonant.In this paper,we show that the maximally resonant{(3,4),4}-fullerene graphs are S_6,S_8,S_(10)^(2),S_(12)^(2),S_(12)^(4),S_(12)^(5),S_(14)^(3),S_(14)^(5),S_(16)^(3),S_(18)^(5),S_(24)as shown in Fig.1.As a corollary,it is shown that if a{(3,4),4}-fullerene graph is 4-resonant,then it is also maximally resonant.
基金Supported by National Natural Science Foundation of China(11801148,11801149 and 11626089)the Foundation for the Doctor of Henan Polytechnic University(B2014-060)
文摘A(3,6)-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons,and has the connectivity 2 or 3.The(3,6)-fullerenes with connectivity 2 are the tubes consisting of l concentric hexagonal layers such that each layer consists of two hexangons,capped on each end by two adjacent triangles,denoted by T_(l)(l≥1).A(3,6)-fullerene Tl with n vertices has exactly 2n/4+1 perfect matchings.The structure of a(3,6)-fullerene G with connectivity 3 can be determined by only three parameters r,s and t,thus we denote it by G=(r,s,t),where r is the radius(number of rings),s is the size(number of spokes in each layer,s(≥4,s is even),and t is the torsion(0≤t<s,t≡r mod 2).In this paper,the counting formula of the perfect matchings in G=n+1,4,t)is given,and the number of perfect matchpings is obtained.Therefore,the correctness of the conclusion that every bridgeless cubic graph with p vertices has at least 2p/3656perfect matchings proposed by Esperet et al is verified for(3,6)-fullerene G=(n+1,4,t).
