Heinz Adolf Jung

On a class of posets and the corresponding comparability graphs

Abstract The concept of an order-theoretical tree is generalized to the notion of a multitree. The comparability graphs of multitrees are characterized and studied with respect to minimal path coverings.

A note on fragments of infinite graphs

Results involving automorphisms and fragments of infinite graphs are proved. In particular for a given fragmentC and a vertex-transitive subgroupG of the automorphism group of a connected graph there exists σ≠G such that σ[C] ⊂C. This proves the countable case of a conjecture of L. Babai and M. E. Watkins concerning graphs allowing a vertex-transitive torsion group action.

Hamiltonism, degree sum and neighborhood intersections

Abstract We give a sufficient condition for hamiltonism of a 2-connected graph involving the degree sum and the neighborhood intersection of any three independent vertices.

Fragments and Automorphisms of Infinite Graphs

Es werden unendliche Graphen untersucht, die sich durch endliche trennende Mengen in mindestens zwei unendliche Teile zerlegen lassen. Dabei wird insbesondere der Frage nachgegangen, wie sich die Zusammenhangsstruktur und die Struktur der Automorphismengruppe wechselseitig beeinflussen.

On the structure of infinite vertex-transitive graphs

Abstract A characterization of all vertex-transitive graphs Γ of connectivity l is given in terms of the lobe structure of Γ. Among these, all graphs are determined whose automorphism groups act primitively (respectively, regularly) on the vertex set.

Longest cycles in tough graphs

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. Akiyama, Exoo, and Harary conjectured that [Δ/2] ≤ la(G) ≤ [Δ+1/2] for any simple graph G with maximum degree Δ. The conjecture has been proved to be true for graphs having Δ = 1, 2, 3, 4, 5, 6, 8, 10. Combining these results, we prove in the article that the conjecture is true for planar graphs having Δ(G) ≠ 7. Several related results assuming some conditions on the girth are obtained as well.

Some results on ends and automorphisms of graphs

Automorphisms ? of a connected graph X satisfying ?(F) ? F for all finite non-empty subsets F of V(X) are of particular interest in the theory of ends. Elementary properties of those automorphisms are studied and linked to the concept of a strip.

On finite fixed sets in infinite graphs

Abstract An automorphism of a graph X is called a translation of X if it fixes no finite non-empty set of vertices of X . It is shown that a group G of automorphisms of the connected graph X fixes a finite non-empty set of vertices or ends of X if and only if any two translations of X in G have a common fixed end. Applications and refinements are discussed.

The connectivities of locally finite primitive graphs

Letγ be an infinite, locally finite graph whose automorphism group is primitive on its vertex set. It is shown that the connectivity ofγ cannot equal 2, but all other values 0, 1, 3, 4, ... are possible.

On 3-skein isomorphisms of graphs

It is shown that a 3-skein isomorphism between 3-connected graphs with at least 5 vertices is induced by an isomorphism. These graphs have no loops but may be infinite and have multiple edges.