General Edge-isoperimetric Inequalities, Part I: Information-theoretical Methods

In combinatorics we often meet two kinds of extremal problems. In one kind , optimal configurations consist of 'objects' , which are somehow uniformly spread in the space under consideration ; and in the other kind , optimal configurations consist of 'objects' , which are somehow compressed. To the first kind belong packing , covering and coding problems , whereas diametric (especially of Erdö s – Ko – Rado type) , vertex-and edge-isoperimetric problems belong to the second kind. For many problems of the spreading type , the probabilistic method gives good or even asymptotically optimal results but , mostly , strictly optimal configurations are unknown. In contrast , problems of the compressing type can often be solved exactly with pushing techniques ('pushing down' , 'pushing to the left' etc. ; see [14]). However , the success of pushing operations is linked to the property that there is a 'nested' structure of optimal configurations with respect to some order. When this is not the case , then there are competing configurations (for example , in [6]) and solutions are harder to obtain. We concentrate here on edge-isoperimetric problems. They can be defined for any graph G ϭ (ᐂ , Ᏹ) as follows. For any A ' ᐂ , define the set Ꮾ (A) of all boundary edges ; that is , Ꮾ (A) ϭ ͕͕ x , y ͖ ෈ Ᏹ : ͉ ͕ x , y ͖ ʝ A ͉ ϭ 1 ͖. (1. 1) P ROBLEM 1. For given positive integer m , find a set A ' ᐂ of cardinality m with minimal possible value of ͉ Ꮾ (A) ͉. A similar problem in this. P ROBLEM 2. For given positive integer m , find a set A ' ᐂ of cardinality m with maximal possible value of ͉ ᏶ (A) ͉ , where ᏶ (A) ϭ ͕͕ x , y ͖ ෈ Ᏹ : ͕ x , y ͖ ' A ͖ is the set of inner edges of A. Notice that , for regular graphs G of degree d , ͉ Ꮾ (A) ͉ ϩ 2 ͉ ᏶ (A) ͉ ϭ d ͉ A ͉ and in this case Problems 1 and 2 are equivalent in the sense that a solution of one of these problems is at the same time a solution of the other. Most results in the literature concern graphs the vertex …