We prove Diestel’s conjecture that the square G of a 2-connected locally finite graph G has a Hamilton circle, a homeomorphic copy of the complex unit circle S in the Freudenthal compactification of G.

We show that the Baire measurable chromatic number of every locally finite Borel graph on a non-empty Polish space is strictly less than twice its ordinary chromatic number, provided this ordinary chromatic number is finite. In the special case that the connectedness equivalence relation is hyperfinite, we obtain the analogous result for the μ-measurable chromatic number.

A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point by repeatedly adding dominated vertices. In finite case, graphs are precisely cop-win graphs, but for infinite situation not well understood. One of our aims in this paper to give a that constructible. This first known such example. We also show every countable ordinal arises as rank some g...

Let $$G=(V,E)$$ be a locally finite, connected and weighted graph. We prove that, for graph satisfying curvature dimension condition $$CDE'(n,0)$$ uniform polynomial volume growth of degree m, all non-negative solutions the equation $$\partial _tu=\Delta u+u^{1+\alpha }$$ blow up in finite time, provided that $$\alpha =\frac{2}{m}$$ . also consider blow-up problem under certain conditions initi...

We show that the deficiency indices of minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to number volume ends. Moreover, we provide criteria, formulated in terms ends, for be closed.

In this work, we study direct limits of finite dimensional basic classical simple Lie superalgebras and obtain the conjugacy classes of Cartan subalgebras under the group of automorphisms.

Abstract We study the structure and compute stable rank of $C^{*}$ -algebras finite higher-rank graphs. completely determine -algebra when $k$ -graph either contains no cycle with an entrance or is cofinal. also exactly which finite, locally convex -graphs yield unital stably -algebras. give several examples to illustrate our results.

We prove a number of auxiliary facts about graphs, mainly about vertex sequences of chains and oriented chains. Then we define a graph to be well-founded if for each vertex in the graph the length of oriented chains ending at the vertex is bounded. A well-founded graph does not have directed cycles or infinite descending chains. In the second part of the article we prove some auxiliary facts ab...

We prove a number of auxiliary facts about graphs, mainly about vertex sequences of chains and oriented chains. Then we define a graph to be well-founded if for each vertex in the graph the length of oriented chains ending at the vertex is bounded. A wellfounded graph does not have directed cycles or infinite descending chains. In the second part of the article we prove some auxiliary facts abo...

This paper explores how to reason locally about global properties of graph transformations, in particular properties involving transitive closure of the edge relations of the graph. We show under which conditions we can soundly reduce reasoning about all nodes in the graph to reasoning about a finite set of nodes. We then give an effective procedure to turn this reduced problem into a Boolean s...