scholarly journals Strongly Maximal Matchings in Infinite Graphs

10.37236/860 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ron Aharoni ◽  
Eli Berger ◽  
Agelos Georgakopoulos ◽  
Philipp Sprüssel
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
Maximal Matching ◽  
Irrational Values

Given an assignment of weights $w$ to the edges of an infinite graph $G$, a matching $M$ in $G$ is called strongly $w$-maximal if for any matching $N$ there holds $\sum\{w(e) \mid e \in N \setminus M\} \le \sum\{w(e) \mid e \in M \setminus N\}$. We prove that if $w$ assumes only finitely many values all of which are rational then $G$ has a strongly $w$-maximal matching. This result is best possible in the sense that if we allow irrational values or infinitely many values then there need not be a strongly $w$-maximal matching.

scholarly journals Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

10.37236/3046 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Simon M. Smith ◽  
Mark E. Watkins
Keyword(s):  
Connected Graph ◽  
Automorphism Groups ◽  
Cardinal Number ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Finite Radius ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Identity Permutation ◽  
Imprimitive Group

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Matroid and Tutte-Connectivity in Infinite Graphs

10.37236/2832 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Henning Bruhn
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
Dual Graphs ◽  
Connectivity Function ◽  
Cycle Matroid ◽  
Infinite Cycles

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.

Menger's Theorem for a Countable Source Set

1994 ◽  
Vol 3 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Ron Aharoni ◽  
Reinhard Diestel
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Menger's Theorem

Paul Erdős has conjectured that Menger's theorem extends to infinite graphs in the following way: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjoint A−B paths and an A−B separator in this graph such that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths to B from all but countably many vertices of A. In particular, the conjecture is true when A is countable.

scholarly journals Functions with finite Dirichlet sum of order p and quasi-monomorphisms of infinite graphs

2012 ◽  
Vol 207 ◽  
pp. 95-138 ◽  
Author(s):  
Tae Hattori ◽  
Atsushi Kasue
Keyword(s):  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Space Form ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Infinite Networks ◽  
Hyperbolic Space Form

AbstractIn this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

scholarly journals An advance in infinite graph models for the analysis of transportation networks

2016 ◽  
Vol 26 (4) ◽  
pp. 855-869
Author(s):  
Martín Cera ◽  
Eugenio M. Fedriani
Keyword(s):  
Graph Theory ◽  
Transportation Networks ◽  
Finite Graph ◽  
Infinite Graph ◽  
Average Degree ◽  
Infinite Graphs ◽  
Graph Models ◽  
Dangerous Goods ◽  
Finite Case ◽  
Percolation Thresholds

Abstract This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.

scholarly journals The Growth of Infinite Graphs: Boundedness and Finite Spreading

1994 ◽  
Vol 3 (1) ◽  
pp. 51-55
Author(s):  
Reinhard Diestel ◽  
Imre Leader
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
Natural Numbers

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.

The Spectrum of an Infinite Graph

2000 ◽  
Vol 52 (5) ◽  
pp. 1057-1084 ◽  
Author(s):  
Hajime Urakawa
Keyword(s):  
Lower Bound ◽  
Essential Spectrum ◽  
Upper Bound ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Discrete Laplacian ◽  
Sharp Estimates

AbstractIn this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.

scholarly journals A STRUCTURED INVERSE SPECTRUM PROBLEM FOR INFINITE GRAPHS AND UNBOUNDED OPERATORS

2018 ◽  
Vol 98 (3) ◽  
pp. 363-371
Author(s):  
EHSSAN KHANMOHAMMADI
Keyword(s):  
Adjoint Operator ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Unbounded Operators ◽  
Real Numbers ◽  
Infinite Set

Let $G$ be an infinite graph on countably many vertices and let $\unicode[STIX]{x1D6EC}$ be a closed, infinite set of real numbers. We establish the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $\unicode[STIX]{x1D6EC}$.

scholarly journals The Distinguishing Index of Infinite Graphs

10.37236/3933 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Izak Broere ◽  
Monika Pilśniak
Keyword(s):  
Random Graph ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Distinguishing Number ◽  
General Bound ◽  
Delta G ◽  
Edge Colouring

The  distinguishing index $D^\prime(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has an edge colouring with $d$ colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined with respect to vertex colourings.We derive several bounds for infinite graphs, in particular, we prove the general bound $D^\prime(G)\leq\Delta(G)$ for an arbitrary infinite graph. Nonetheless,  the distinguishing index is at most two for many countable graphs, also for the infinite random graph and for uncountable tree-like graphs.We also investigate the concept of the motion of edges and its relationship with the Infinite Motion Lemma. 

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