An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph We prove an old conjecture of Halin, which characterizes the bounded graphs in terms of four forbidden topological subgraphs

/pdf/a-proof-of-the-bounded-graph-conjecture-4ojdhrbk0q.pdf

A proof of the bounded graph conjecture.

On spanning trees and k -connectedness in infinite graphs

An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. Thomassen has conjectured that a countable graph is bounded if and only if its edges can be oriented, possibly both ways, so that every vertex has finite out-degree and every ray has a forward oriented tail. We present a counterexample to this conjecture.

/pdf/the-growth-of-infinite-graphs-boundedness-and-finite-10wc5j191t.pdf

The Growth of Infinite Graphs: Boundedness and Finite Spreading

For any graph $G$, the First-Fit (or Grundy) chromatic number of $G$, denoted by $\chi_{_{\sf FF}}(G)$, is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. We call a family $\mathcal{F}$ of graphs $(\delta, \chi_{_{\sf FF}})$-bounded if there exists a function $f(x)$ with $f(x)\rightarrow \infty$ as $x\rightarrow \infty$ such that for any graph $G$ from the family one has $\chi_{_{\sf FF}}(G)\geq f(\delta(G))$, where $\delta(G)$ is the minimum degree of $G$. We first give some results concerning $(\delta, \chi_{_{\sf FF}})$-bounded families and obtain a few such families. Then we prove that for any positive integer $\ell$, $Forb(K_{\ell,\ell})$ is $(\delta, \chi_{_{\sf FF}})$-bounded, where $K_{\ell,\ell}$ is complete bipartite graph. We conjecture that if $G$ is any $C_4$-free graph then $\chi_{_{\sf FF}}(G)\geq \delta(G)+1$. We prove the validity of this conjecture for chordal graphs, complement of bipartite graphs and graphs with low minimum degree.

$(\delta, \chi_{_{\sf FF}})$-bounded families of graphs

An infinite graph G is called dominating if its vertices can be labelled with natural numbers in such a way that for every function f : ω→ ω there is a ray in G whose sequence of labels eventually exceeds f . Conversely, G is called bounded if for every labelling of its vertices with natural numbers there exists a function f : ω→ω which eventually exceeds the labelling along any ray in G. This expository paper describes recent classifications of the dominating and the bounded graphs by forbidden topological minors, and indicates some connections of these results to infinite games.

/pdf/dominating-functions-and-topological-graph-minors-38uy6qt3io.pdf

Dominating functions and topological graph minors.

A graph G is called bounded if for every mapping ?:V(G)?N there exists a sequence (Cn)n?N of naturals such that for every ray U=(u1,u2,u3,?) of G an n0 can be found with ?f(ui)< ci for all i?n0. All locally finite connected graphs are bounded. Four basic types of unbounded graphs are given, and it is conjectured that one of these configurations must be present in every unbounded graph. This conjecture is proved for the graphs with rayless blocks and the graphs which do not contain an infinite system of disjoint rays.

Bounded graphs

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Bounded graphs