Florian Lehner

Distinguishing graphs with intermediate growth

AbstractA graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finte graph with infinite nite motion and growth at most

Distinguishing graphs with infinite motion and nonlinear growth

\mathcal{O}\left( {2^{(1 - \varepsilon )\tfrac{{\sqrt n }} {2}} } \right)

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.

A counterexample to the reconstruction conjecture for locally finite trees

We prove a refinement of the tree packing theorem by Tutte/Nash-Williams for finite graphs. This result is used to obtain a similar result for end faithful spanning tree packings in certain infinite graphs and consequently to establish a sufficient Hamiltonicity condition for the line graphs of such graphs.

Fast Factorization of Cartesian products of (Directed) Hypergraphs

The distinguishing number D( G ) of a graph G is the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs G with infinite motion and growth o ( n 2 / log 2 n ) is either 1 or 2 , which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs G of arbitrary cardinality are 2 -distinguishable if every nontrivial automorphism moves at least uncountably many vertices m( G ) , where m( G ) ≥ ∣Aut( G )∣ . This extends a result of Imrich et al. to graphs with automorphism groups of arbitrary cardinality.

The Cartesian product of graphs with loops

Two graphs

Maximizing the Number of Independent Sets of Fixed Size in Connected Graphs with Given Independence Number

G

Pursuit evasion on infinite graphs

and