Maya Stein

On end degrees and infinite cycles in locally finite graphs

We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.

Connectivity and tree structure in finite graphs

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditionsunder which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph.As an application, we show that the k-blocks — the maximal vertex sets that cannot be separated by at most k vertices — of a graph G live in distinct parts of a suitable treedecomposition of G of adhesion at most k, whose decomposition tree is invariant under the automorphisms of G. This extends recent work of Dunwoody and Krön and, like theirs, generalizes a similar theorem of Tutte for k=2.Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all k simultaneously, all the k-blocks of a finite graph.

Duality of ends

We investigate the end spaces of infinite dual graphs. We show that there exists a natural homeomorphism * between the end spaces of a graph and its dual, and that * preserves the ‘end degree’. In particular, * maps thick ends to thick ends. Along the way, we prove that Tutte-connectivity is invariant under taking (infinite) duals.

The Approximate Loebl--Komlós--Sós Conjecture III: The Finer Structure of LKS Graphs

This is the third of a series of four papers in which we prove the following relaxation of the Loebl--Komlos--Sos conjecture: For every

The Approximate Loebl--Komlós--Sós Conjecture II: The Rough Structure of LKS Graphs

\alpha>0

The Approximate Loebl--Komlós--Sós Conjecture IV: Embedding Techniques and the Proof of the Main Result

there exists a number

The approximate Loebl-Komlos-Sos Conjecture I: The sparse decomposition

k_0

On claw-free t -perfect graphs

such that for every

Extremal infinite graph theory

k>k_0

Monochromatic Cycle Partitions in Local Edge Colorings

, every