scholarly journals Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample

10.37236/6773 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Karl Heuer
Keyword(s):  
Finite Graph ◽  
Infinite Graphs ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Locally Finite ◽  
Finite Graphs ◽  
The Third ◽  
Locally Finite Graph ◽  
Locally Finite Graphs ◽  
A Minor

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.

scholarly journals Geodetic Topological Cycles in Locally Finite Graphs

10.37236/233 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Agelos Georgakopoulos ◽  
Philipp Sprüssel
Keyword(s):  
Finite Graph ◽  
Cycle Space ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graph ◽  
Locally Finite Graphs

We prove that the topological cycle space ${\cal C}(G)$ of a locally finite graph $G$ is generated by its geodetic topological circles. We further show that, although the finite cycles of $G$ generate ${\cal C}(G)$, its finite geodetic cycles need not generate ${\cal C}(G)$.

Random Colourings and Automorphism Breaking in Locally Finite Graphs

2013 ◽  
Vol 22 (6) ◽  
pp. 885-909 ◽  
Author(s):  
FLORIAN LEHNER
Keyword(s):  
Automorphism Group ◽  
Haar Measure ◽  
Finite Graph ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graph ◽  
Locally Finite Graphs ◽  
Null Set

A colouring of a graphGis called distinguishing if its stabilizer in AutGis trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabilizer of such a colouring is almost surely nowhere dense in AutGand a null set with respect to the Haar measure on the automorphism group. We also investigate random 2-colourings in several classes of locally finite graphs where the existence of a distinguishing 2-colouring has already been established. It turns out that in all of these cases a random 2-colouring is almost surely distinguishing.

scholarly journals Orthogonality and Minimality in the Homology of Locally Finite Graphs

10.37236/3844 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Reinhard Diestel ◽  
Julian Pott
Keyword(s):  
Vector Space ◽  
Minimal Element ◽  
Finite Graph ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graph ◽  
Finite Set ◽  
Locally Finite Graphs

Given a finite set $E$, a subset $D\subseteq E$ (viewed as a function $E\to \mathbb F_2$) is orthogonal to a given subspace $\mathcal F$ of the $\mathbb F_2$-vector space of functions $E\to \mathbb F_2$ as soon as $D$ is orthogonal to every $\subseteq$-minimal element of $\mathcal F$. This fails in general when $E$ is infinite.However, we prove the above statement for the six subspaces $\mathcal F$ of the edge space of any $3$-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of Diestel (2010, arXiv:0912.4213).

STABILIZERS OF TRANSITIVE ACTIONS ON LOCALLY FINITE GRAPHS

2000 ◽  
Vol 10 (05) ◽  
pp. 591-602 ◽  
Author(s):  
VOLODYMYR NEKRASHEVYCH
Keyword(s):  
Finite Graph ◽  
Transitive Action ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graph ◽  
Locally Finite Graphs ◽  
Transitive Actions

We prove a criterion for a group to be representable as a vertex stabilizer of a transitive action on a locally finite graph. We show that actions with finite vertex stabilizers are open in a natural topology.

scholarly journals Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs

10.37236/6083 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Johannes Carmesin
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Vertex Sets ◽  
Special Case

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.

Matchings in Arbitrary Graphs

1971 ◽  
Vol 69 (3) ◽  
pp. 401-407 ◽  
Author(s):  
R. A. Brualdi
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Maximum Cardinality ◽  
Original Proof ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

scholarly journals Forcing Finite Minors in Sparse Infinite Graphs by Large-Degree Assumptions

10.37236/2891 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Reinhard Diestel
Keyword(s):  
Large Degree ◽  
Finite Graph ◽  
Average Degree ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Extremal Theory

Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum or average degree. This is part of a wider project which seeks to develop an extremal theory of sparse infinite graphs.

Matchings in Countable Graphs

1977 ◽  
Vol 29 (1) ◽  
pp. 165-168 ◽  
Author(s):  
K. Steffens
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

Tutte [9] has given necessary and sufficient conditions for a finite graph to have a perfect matching. Different proofs are given by Brualdi [1] and Gallai [2; 3]. The shortest proof of Tutte's theorem is due to Lovasz [5]. In another paper [10] Tutte extended his conditions for a perfect matching to locally finite graphs. In [4] Kaluza gave a condition on arbitrary graphs which is entirely different from Tutte's.

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