点度和面度的最小值是3的连通平图

Connected Plane Graphs with min(δ_v,δ_f)=3

称一个连通平图是k||δ_(v,f^-)平图,若其顶点的最小度δ_v和面的最小度δ_f的最小值δ_(v,f)是k.本文研究3||δ_(v,f^-)平图.通过一个图运算构造证明链环分支数等于1的3||δ_(v,f^-)平图的存在性,并证明在相等意义下链环分支数不小于基圈数的3||δ_(v,f^-)平图是唯一的.然后证明在相等意义下,边数等于6,8的3||δ_(v,f^-)平图都是唯一的,边数等于9的3||δ_(v,f^-)平图有且只有两个且它们是互为对偶的.接着研究连通平图与其中间图在相等意义下的相互关系.作为运用,证明了无弓形链环图的三个唯一性结论.

A connected plane graph G is called a k || δv,f-plane graph if δv,f=k.there,δv,f is the minimum value of δv,and δ,f,δv is the minimum degree of vertices of G and δf is the minimum degree of faces of G.We mainly study the 3||δv,f-plane graphs.We first prove the existence of the 3 || δv,f-plane graphs with the link component number 1 by constructing them via a graph operation,and prove the uniqueness of the3||δv,f-plane graph with link component number not less than nullity in the sense of equivalence.Then we prove the uniqueness of 3 || δv,f-plane graph with the edge number 6 and 8 in the sense of equivalence.We also show that there are only two3|| δv,f-plane graphs with the edge number 9 in the sense of equivalence,furthermore,they are dual.After that,we study the correlations between a connected plane graph and its medial graphs in the sense of equivalence.Finally,as applications,we prove three uniqueness conclusions of lune-free link graphs.