Strengthening wilderness protection in Antarctica through lessons learnt from the Arctic : An introduction into a comparative research on regulating tourism and other non-governmental activities between both polar regions (original) (raw)

A Note on Vertex-Disjoint Cycles

Combinatorics, Probability and Computing, 2002

Häggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k2+19k+10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k2 cannot be decreased and, in this sense, this result is best possible.

Existence of two disjoint long cycles in graphs

Discrete Mathematics, 2005

Let n, h be integers with n ≥ 6 and h ≥ 7. We prove that if G is a graph of order n with σ 2 (G) ≥ h, then G contains two disjoint cycles C 1 and C 2 such that |V (C 1)| + |V (C 2)| ≥ min{h, n}.

A Refinement of Theorems on Vertex-Disjoint Chorded Cycles

Graphs and Combinatorics, 2016

In 1963, Corrádi and Hajnal settled a conjecture of Erdős by proving that, for all k ≥ 1, any graph G with |G| ≥ 3k and minimum degree at least 2k contains k vertex-disjoint cycles. In 2008, Finkel proved that for all k ≥ 1, any graph G with |G| ≥ 4k and minimum degree at least 3k contains k vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all k ≥ 1, any graph G with |G| ≥ 4k and minimum Ore-degree at least 6k − 1 contains k vertex-disjoint cycles. We refine this result, characterizing the graphs G with |G| ≥ 4k and minimum Ore-degree at least 6k − 2 that do not have k disjoint chorded cycles.

On the Number of Cycles in a Graph with Restricted Cycle Lengths

SIAM Journal on Discrete Mathematics, 2018

Let L be a set of positive integers. We call a (directed) graph G an L-cycle graph if all cycle lengths in G belong to L. Let c(L, n) be the maximum number of cycles possible in an n-vertex L-cycle graph (we use c(L, n) for the number of cycles in directed graphs). In the undirected case we show that for any fixed set L, we have c(L, n) = Θ(n k/ ) where k is the largest element of L and 2 is the smallest even element of L (if L contains only odd elements, then c(L, n) = Θ(n) holds.) We also give a characterization of L-cycle graphs when L is a single element. In the directed case we prove that for any fixed set L we have c(L, n) = (1 + o( ))( n-1 k-1 ) k-1 , where k is the largest element of L. We determine the exact value of c({k}, n) for every k and characterize all graphs attaining this maximum.

Polarities and 2k-cycle-free graphs

Discrete Mathematics, 1999

Let C2, be the cycle on 2k vertices, and let ex(v, C2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C2,. In this paper we discuss a method which allows one to sometimes improve numerical constants in lower bounds for ex(v, C:k). The method utilizes polarities in certain rank two geometries. It is applied to refute some conjectures about the values of ex(v, C2k), and to construct some new examples of graphs having certain restrictions on the lengths of their cycles. In particular, we construct an infinite family {Gi} of Cr-free graphs with IE(G,)[ ~/IV(G,)I 4'3, i-+ ec, which improves the constant in the previous best lower bound on ex(v, Cr) from 2/34'3 ~0.462 to 1/2.

On certain cycles in graphs

Proceedings of the Edinburgh Mathematical Society, 1981

We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.

Disjoint chorded cycles in graphs

Discrete Mathematics, 2008

We propose the following conjecture to generalize results of Pósa and of Corrádi and Hajnal. Let r, s be nonnegative integers and let G be a graph with |V (G)| ≥ 3r + 4s and minimal degree δ(G) ≥ 2r + 3s. Then G contains a collection of r + s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r = 0, s = 2 and for s = 1. The corresponding extremal problem, to find the minimum number of edges in a graph on n vertices ensuring the existence of two vertex disjoint chorded cycles, is also settled.

Challenges in Protecting the Wilderness of Antarctica

Since 1998, the wilderness values of Antarctica have been among those given legal recognition under the Protocol on Environmental Protection to the Antarctic Treaty. Despite the legal obligation, on-the-ground implementation has attracted little interest. The term “wilderness” and its consequential operational implication, including the designation of Antarctic Specially Protected Areas and the drafting of Environmental Impact Assessments, is still poorly conceptualized in Antarctic Treaty System discourse. Many possible factors underlie the lack of attention to the protection of wilderness in Antarctica. There is the perception that wilderness is in overabundance in Antarctica and hence does not require special protection. Setting areas aside, out of bounds of infrastructure development, may be perceived as threatening to national ambitions and the accepted ideas of freedom of movement. There is no formal definition of either term in the Protocol or elsewhere in Antarctic Treaty System (ATS) instruments, and the concept of wilderness (as other terms in the Protocol) seems often to be cast as too complex or philosophical to be applied in practice. We ask the question of how existing environmental measures within the ATS and non-Antarctic wilderness management tools could be used to achieve on-the-ground protection of the Antarctic wilderness.