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Looseness and Independence Number of Triangulations on Closed Surfaces

100%

Nakamoto A., Negami S., Ohba K., Suzuki Y.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 3

545-554

The looseness of a triangulation G on a closed surface F2, denoted by ξ (G), is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , k + 3}, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound ξ (G) for triangulations G on closed surfaces by the independence number of G denoted by α(G). In particular, for a triangulation G on the sphere, we have [...] and this bound is sharp. For a triangulation G on a non-spherical surface F2, we have ξ (G) ≤ 2α(G) + l(F2) − 2, where l(F2) = [(2 − χ(F2))/2] with Euler characteristic χ(F2).

The minimizing of the Nielsen root classes

72%

Gonçalves D., Aniz C.

Open Mathematics

2004

tom 2

nr 1

112-122

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M 2)−χ(M 10/(1−χ(M 2)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M 2)−χ(M 2)/(1−χ(M 2)). Also we construct, for each integer n≥3, an example of a map f: K n→N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Open Mathematics

1 Aniz C.

1 Gonçalves D.

1 Nakamoto A.

1 Negami S.

1 Ohba K.

1 Suzuki Y.

1 2016

1 2004

Wyniki wyszukiwania

Looseness and Independence Number of Triangulations on Closed Surfaces

The minimizing of the Nielsen root classes