In his recent collection [ 2 ] of open problems about infinite graphs, Halin defines the following hierarchy of graphs. He defines Γ0 to be the class of all graphs (finite or infinite), and for each ordinal μ > 0 he defines Γμ as the class of all graphs containing, for each λ < μ, infinitely many disjoint connected graphs from Γλ. Thus, Γ1 is the class of infinite graphs, Γ2 is the class of all...

Halin conjectured 20 years ago that a graph has normal spanning tree if and only every minor of it countable colouring number. We prove Halin’s conjecture. This implies forbidden characterisation for the property having tree.

In this paper we study the cycle base structures of embedded graphs on surfaces. We first give a sufficient and necessary condition for a set of facial cycles to be contained in a minimum cycle base (or MCB in short) and then set up a 1-1 correspondence between the set of MCBs and the set of collections of nonseparating cycles which are in general positions on surfaces and are of shortest total...

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to ...

A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Hahn graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin g...

Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the following two conditions: (i) G contains no subdivision of a ‘fat’ Kא0 , one in which every edge has been replaced by uncountably many parallel edges; and (ii) G has no Kא0 subgraph. We show that the second condition is unnecessary.

Ozonolysis is a major removal mechanism in the troposphere for unsaturated hydrocarbons which are emitted in large quantities from both natural and human sources. Now it is generally accepted that ozonolysis of alkenes proceeds via Criegee intermediates, highly reactive species postulated in 1949 by Rudolf Criegee. In the troposphere, Criegee intermediates are involved in several important atmo...

Florian M.E. Wagenlehner, Johny E. Elkahwaji*, Ferran Algaba†, Truls Bjerklund-Johansen‡, Kurt G. Naber¶, Rudolf Hartung§ and Wolfgang Weidner Department of Urology, Justus-Liebig-University Giessen, Germany, *Department of Surgery, Division of Urological Surgery, University of Nebraska Medical Center, Omaha, USA, †Department of Pathology, Fundacio Puigvert, Barcelona, Spain, ‡Department of Uro...