Herbert Fleischner

The square of every two-connected graph is Hamiltonian

Abstract A graph G is a line-critical block if κ(G) = 2 and if for any line e of G the graph G − e has κ(G − e) = 1. If G is a line-critical block, then G is either a DT-block (i.e., G is a two-connected graph in which every line is incident to a point of degree two), or G contains a specific two-connected subgraph which is a DT-block (Theorem 1). Using this result and results of the preceding paper on DT-graphs, a simple proof of the conjecture that the square of every two-connected graph is Hamiltonian is given.

Eine gemeinsame Basis für die Theorie der Eulerschen Graphen und den Satz von Petersen

The main result states: Lete1,e2,e3 be three lines incident to the pointv (degv≥4) of the connected bridgeless graphG such thate1 ande3 belong to different blocks ifv is a cutpoint. “Split the pointv” in two ways: Lete1,ej,j=2, 3, be incident to a new pointv1j and leave the remainder ofG unchanged, thus obtainingG1j. Then at least one of the two graphsG12,G13 is connected and bridgeless. — A classical result ofFrink follows from this theorem which is the key to a simple proof of Petersens theorem. Moreover, the above result can be used to prove practically all classical results on Eulerian graphs, including best upper and lower bounds for the number of Eulerian trails in a connected Eulerian graph. In the theory of Eulerian graphs, it can be viewed as the basis for good algorithms checking on several properties of this class of graphs.

On spanning subgraphs of a connected bridgeless graph and their application to DT-graphs

The graphs considered are connected and bridgeless. For such graphs the existence of two types of connected spanning subgraphs is proved. Applying these results to a connected bridgeless DT-graph (i.e., every line is incident to a point of degree two), G, one obtains the existence of specific Hamiltonian cycles and Hamiltonian paths in G2. In addition it is proved that the square of a connected bridgeless DT-graph is Hamiltonian connected.

Gedanken zur Vier-Farben-Vermutung

It is shown that the four colour conjecture is equivalent to a conjecture on the existence of spanning Eulerian subgraphs of a certain type in triangulations of the plane. From this, a simple proof of Heawoods Equivalence Theorem is given. Moreover, an elementary operation is defined for triangulations of the plane; and by repeated application of this operation one can construct a given colouring from another given colouring.