Research On The Functional Equations Of Unicursal Maps

In this thesis, we study the functional equations of unicursal maps on the basis of eulerian maps. Eulerian maps are one type of important maps and have played a crucial role in enumerating maps. In particular, Tutte’s sum-free formula for the number of eulerian planar maps, all of whose vertices are labelled with ends distinguished on the basis of vertex partition given in obtaining his ground-breaking formula for counting rooted planar maps by the number of edges.Y.P.Liu[32] discussed the enumeration of rooted loopless planar maps and es-tablished its functional equation satisfied by the enufunction with the valency of root-vertex and the number of the vertex partition as parameters; and investigated eulerian loopless planar maps and simple maps on the basis of the result of loop-less maps; Y.P.Liu[33] provided the functional equation of simple maps with the valency of root-face and the number of the face partition as parameters; J.L.Cai and Y.P.Liu[59] have ever got a complicated functional equation of rooted simple eulerian planar maps with the valency of root-vertex and the number of the in-ner edges and the valency of root-face as parameters, but there is still no explicit formula of simple Eulerian maps. But during the process of investigating simple Eulerian planar maps, we made a series of study of unicursal maps.Liskovets and Walsh(2004) investigated one type of planar maps:unicursal maps-maps with exactly two vertices of odd valency and first obtained a sum-free formula for the number of rooted unicursal planar maps with a given number of edges. In that paper, they had also found a sum-free formula for the number of unicursal maps rooted in a vertex of odd valency and a formula for the number of rooted unicursal maps as a function of the odd vertex valencies.Y.P.Liu[81] investigated the enumeration of outerplanar triangulations, planar triangulations, triangulations on the disk, triangulations on the projective plane and triangulations on the torus respectively, and established a form of functional equation satisfied by the enufunction of rooted Eulerian near-triangulations with the valency of root-face and the number of non-rooted faces as parameters; and also the enumeration of Eulerian maps on surface, and got its functional equation.In this paper we investigate the enumerative problems of four types of rooted planar maps:loopless unicursal planar maps, unicursal planar near-triangulations, unicursal planar near-quadrangulations and unicursal maps on surface. There is no result about the planar maps mentioned above as we known up to now, except the results here. In the following, We will introduce the content of each chapter of this thesis briefly.In Chapter1, firstly, we give a brief introduction on concepts and background of unicursal maps. Secondly, we introduce some terminologies on graph theoryIn Chapter2, we study a functional equation satisfied by the generating func-tion for enumerating rooted loopless unicursal planar maps with the valency of the root-vertex and the vertex partition as parameters.In Chapter3, we use the enufunction approach to enumerate rooted unicursal planar near-triangulations with the valency of the root-face and the number of non-rooted faces as parameters.In Chapter4, we will decompose Eulerian planar near-quadrangulations and provide functional equations of the enufunction of rooted Eulerian planar near-quadrangulations with the valency of the root-face and the number of non-rooted faces as parameters,In Chapter5, we use the enufunction approach to enumerate rooted unicursal planar near-quadrangulations with the valency of the root-face and the number of non-rooted faces as parametersIn Chapter6, the main result in this paper is to establish a form of func-tional equation satisfied by enumerating function of unicursal maps on surfaces. The parameters of this function are the valency of the root-vertex, the number of nonrooted vertices of valency i.In Chapter7, we posed some questions for further study.