Pago del impuesto a la renta

ξ(G) = X

π_∈Π(G)

ξ(Ωπ)

is the number of embeddings of G on the surface of relative genus k,

−β(G) ≤k ≤ ⌊β(₂G)⌋.

A graph of order 2 without selfloop is called a link bundle.

E1.13 Let Lm be the link bundle of size m ≥ 1. Determine

ξ(Lm).

A graph of order 1 is also called a bouquet, or a loop bundle. E1.14 Let Bm be a bouquet of size m, m ≥ 1. Determine

ξ(Bm).

A graph of order 2 is called a bipole. Of course, a link bundle is a bipole which has no selfloop.

E1.15 Let Pm be a bipole of size m, m ≥ 1. Determine ξ(Pm).

I.8 Researches

The set(repetition at most twice of elements permitted) of edges appearing on a travel can be shown to have a partition of each subset forming still a subtravel except probably the travel itself. Such a partition is called adecompositionof a travel into subtravels. However, it is not yet known if any travel can be decomposed into tours except only the case that its induced graph has a cut-edge.

R1.1 Prove, or disprove, the conjecture that a travel with at most twice occurrences of an edge in a graph has a decomposition into tours if, and only if, the induced subgraph of the travel is without cut-edge.

Because a circuit is restricted from a tour by no repetition of a vertex, the following conjecture would look stronger the last one.

most twice occurrences of an edge in a graph has a decomposition into circuits if, and only if, the induced subgraph of the travel is without cut-edge.

However, it can be shown from Theorem 1.3 that any tour has a decomposition into circuits. The above two conjectures are, in fact, equivalent. Because a cut-edge is never on a circuit, the necessity is always true. A travel with three occurrences of an edge permitted does not have a decomposition into circuits in general. For example, on the graph determined by Par= _{{x(0), y(0)_},_{x(1), y(1)_}}, the travel

xx−1xy−1 where x = _hx(0), x(1)_i and y = _hy(0), y(1)_i has no circuit decomposition.

Furthermore, the two conjectures are apparently right when the graph is planar because each face boundary of its planar embedding is generally a tour whenever without cut-edge.

R1.3 For a given graph G and an integer p, p _≥ 0, find the number np(G) of embeddings of G on the orientable surface of genus

p.

The aim is at the genus distribution of embeddings of G on ori- entable surfaces, i.e., the polynomial

PO(G) =

⌊σ/2_⌋

X

p=0

np(G)xp,

where σ is the Betti number of G.

For p = 0, n0(G) can be done based on [Liu6]. If G is planar,

O1.11 provides the result for 2-connected case. Others can also be derived. As to justify if a graph is planar, a theory can be seen from Chapters 3,5 and 7 in [Liu5].

Generally speaking, not easy to get the complete answer in a short period of time. However, the following approach would be avail- able to access this problem. Choose a special type of graphs, for instance, a wheel(a circuit Cn all of whose vertices are adjacent to an

I.8 Researches 39 except for all articulate vertices forming the vertex set of the circuit) and so forth.

Of course, the technique and theoretical results in 1.3 can be employed to calculate the number of distinct embeddings of a graph by hand and by computer.

R1.4 Orientable single peak conjecture. The coefficients of the polynomial in R1.3 are of single peak , i.e., they are from increase to decrease as p runs from 0 to ⌊σ/2⌋(≥ 3), np(G).

The purpose here is to prove, or disprove the conjecture not nec- essary to get all np(G), 0≤ p ≤ ⌊σ/2⌋(≥3).

R1.5 Determine the number of distinct embeddings, which have one, or two faces, of a graph on orientable surfaces.

R1.6 For a given graphGand an integerq, q _≥1, find the num- ber ˜nq(G) of distinct embeddings on nonorientable surfaces of genus

q.

The aim is at the genus polynomial of embeddings ofGon nonori- entable surfaces: PN(G) = σ X q=1 ˜ nq(G)xq,

where σ is the Betti number of G.

Some pre-investigations for G is that a wheel, or a generalized Halin graph can firstly be done.

R1.7 For a graphG, justify if it is embeddable on the projective plane, and then determine ˜n1(G) according to the connectivity of G.

R1.8For a graph embeddable on the projective plane, determine how many sets of circuits such that for each, all of its circuits are essential if, and only if, one of them is essential in an embedding of G

on the projective plane.

R1.9 Nonorientable single peak conjecture. The coefficients of the polynomial in R1.6 are of single peak in the interval [o, σ] where

R1.10 For a given type of graphs _G and an integer p, find the number of distinct embeddings of graphs inG on the orientable surface of genus p. Further, determine the polynomial

PO(G) =

⌊σ_X(G)/2⌋

p=0

np(G)xp

where σ(_G) = max_{σ(G)_|G _{∈ G}}.

R1.11 For a given type of graphs G and an integer q, q ≥ 1, find the number of embeddings of graphs in G on the nonorientable surface of genus q. Further, determine the polynomial

PN(G) = σ(_G) X q=1 ˜ nq(G)xq where σ(_G) = max_{σ(G)_|G _{∈ G}}.

R1.12 For a set of graphs with some fixed invariants, extract sharp bounds(lower or upper) of the orientable minimum genus and sharp bounds(lower or upper) of orientable maximum genus.

Here, invariants are chosen from theorder(vertex number),size(edge number), chromatic number (the minimum number of colors by which vertices of a graph can be colored such that adjacent vertices have distinct colors), crossing number(the minimum number of crossing in- ner points among all planar immersions of a graph), thickness (the minimum number of subsets among all partitions of the edge set such that each of the subsets induces a planar graph), and so forth.

R1.13 For a set of graphs and a set of invariants fixed, pro- vide sharp bounds(lower or upper) of minimum nonorientable genus of embeddings of graphs in the set.