By Al-Jaber Ah.

A number of points concerning the combinatorial homes of heapsort are mentioned during this thesis. A recursion formulation for the variety of lots pleasing a given situation among any offsprings with a similar mum or dad Is given and a number of other houses of lots are mentioned together with a brand new set of rules to generate the set of all tons of any dimension. additionally during this paintings we outline moment order bushes that have an exceptional value within the research of the complexity of Williams' algorithms to generate a heap. We talk about this type of bushes and we end up that the producing functionality of the variety of bushes satisfies a nonlinear differential distinction equation. The numerical computation and the asymptotic enlargement for a volume relating to this nonlinear differential distinction equation Is given during this paintings . eventually, we supply an higher sure for the variety of the second one order timber generated from the set of all tons of measurement N the place N has the shape 2-1 for any confident integer ok.

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A' .. ",. • ~.................. ~...... • _ ..... 9. I ;; ••• • Y d) ... I ... "" \... :, \..... ". : \ ..... ',. ~--(' Z W I : • •.... -- I r. ,' . " . r - •• ..... --....... : .... ,,' . • ..... ' I b) I ............ --,..... ". , I 47 ............. '. • ................ - ......... 48 3. Classification of Surfaces The above process transforms U into another unitary gem U'. The handle {X, Y} has been replaced by the assembled handle {W, Z}. The red edges in any assembled cross-cap or in a member of any other assembled handle are in These cross-caps and handles therefore remain assembled.

31 -n~ '. 8. It may also be true that the red-blue bigons of a 3-graph are all squares. In this case, the 3-graph is called a gem. ] The red-blue bigons of a gem sometimes are called bisquares. Let K be a gem (H, V, F). Every red-blue bigon has two red edges, each of which belongs to a unique red-yellow bigon. Thus, each bisquare meets just two red-yellow bigons, which may not be distinct. Let G(K) be the graph in which VG(K) is the set of red-yellow bigons, EG(K) is the set of red-blue bigons, and each edge joins the vertices it meets.

Thus, non-empty connected planar graphs are the graphs imbeddable in the sphere. They are the ones that underlie spherical gems. In general, the determination of the genus of a graph is an unsolved problem. However, a number of characterisations of planar graphs are known, and we describe several in this book. The reader is referred to Gross and Tucker (1987) for an account of techniques used to find the genus of a graph. 3 and the fact that IV K I = 4IY(K)1 for any gem K. 9 Let K be a connected gem.