Local Topological Toughness and Local Factors

We localize and strengthen Katona’s idea of an edge-toughness to a local topological toughness. We disprove a conjecture of Katona concerning the conection between edge-toughness and factors. For the topological toughness we prove a theorem similar to Katona’s 2k-factorconjecture, which turned out to be false for his edge-toughness. We prove, that besides this the topological toughness has nearly all known nice properties of Katona’s edge-toughness and therefore is worth to be considered. 1. Preliminaries and Results For notations not defined here we refer to [2]. Unless otherwise stated, t is an arbitrary non negative real number, k is an arbitrary integer, G is an arbitrary finite graph (loops and multiple edges allowed), U is an arbitrary subgraph of G, X and H are arbitrary disjoint subsets of V (G), Y is an arbitrary subset of E(G−X−H), and f is an arbitrary function that maps H into the positive integers. An H-path is a path connecting two different vertices of H. A cycle covering H is called an H-cycle. The union of internally disjoint H-paths is called an H-local k-factor, if all vertices of H have degree k in it, a partial H-local k-factor, if all vertices of H have at most degree k in it, an H-local f -factor if each vertex h of H has degree f(h) in it, and a partial H-local f -factor if each vertex h of H has at most degree f(h) in it. The size of H-local factors is the number of its Hpaths. The maximum number of internally disjoint H-paths is denoted by pG(H). With G[X] we denote the subgraph of G induced by X, [Y ] denotes the graph with edge set Y whose vertex set is the set of all vertices incident with edges of Y . Instead of G[V ([Y ])] we shortly write G[Y ]. E ′(G) denotes the set of all edges in G except the loops. Let C(G) denote the set of components of G and ∂G(U) denote the set of vertices of U incident with ∗ Research supported by the “Mathematics in Information Society” project carried out by Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences, in the framework of the European Community’s “Confirming the International Role of Community Research” programme. e-mail: [email protected] † Research supported by the Ministry of Education OTKA grant OTKA T 043520. e-mail: [email protected] 2 Frank Göring, Gyula Y. Katona edges of G−E(U). For V (U)− ∂G(U) we will write shortly inG(U). According to [10] we define the permeability of a pair (X, Y ) by: permG(X, Y ) = |X|+ ∑ C∈C([Y ]) ⌊ |∂G−X(C)| 2 ⌋ The following definitions generalize this concept: Let G be a graph, and f be a function mapping H∗ ⊆ V (G) into the set of positive integers. An f -separator of G is a pair (X, Y ) with X ⊆ V (G), Y ⊆ E(G−X) and ∂G−XY disjoint to H∗ such that G−X − Y has no H∗-paths. The permeability of an f -separator is permG,f (X, Y ) = |X \H∗|+ ∑