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.

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 Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

Asymptotic Behavior of Skew Conditional Heat Kernels on Graph Networks

1993 ◽  
Vol 45 (4) ◽  
pp. 863-878 ◽  
Author(s):  
Tatsuya Okada
Keyword(s):  
Asymptotic Behavior ◽  
Finite Graph ◽  
Heat Propagation ◽  
Heat Kernels ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Periodic Models ◽  
Definition Of

AbstractIn this note, we will consider the heat propagation on locally finite graph networks which satisfy a skew condition on vertices (See Definition of Section 2). For several periodic models, we will construct the heat kernels Pt with the skew condition explicitly, and derive the decay order of Pt as time goes to infinity.

scholarly journals The homology of a locally finite graph with ends

COMBINATORICA ◽  
2010 ◽  
Vol 30 (6) ◽  
pp. 681-714 ◽  
Author(s):  
Reinhard Diestel ◽  
Philipp Sprüssel
Keyword(s):  
Finite Graph ◽  
Locally Finite ◽  
Locally Finite Graph

scholarly journals The Planarity Theorems of MacLane and Whitney for Graph-like Continua

10.37236/284 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Robin Christian ◽  
R. Bruce Richter ◽  
Brendan Rooney
Keyword(s):  
Finite Graph ◽  
Disjoint Paths ◽  
Locally Finite ◽  
Edge Disjoint ◽  
Locally Finite Graph ◽  
Compact Graph

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane's Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

scholarly journals Relating Different Cycle Spaces of the Same Infinite Graph

10.37236/622 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
R. Bruce Richter ◽  
Brendan Rooney
Keyword(s):  
Finite Graph ◽  
Infinite Graph ◽  
Cycle Space ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Continuous Surjection

Casteels and Richter have shown that if $X$ and $Y$ are distinct compactifications of a locally finite graph $G$ and $f:X\to Y$ is a continuous surjection such that $f$ restricts to a homeomorphism on $G$, then the cycle space $Z_X$ of $X$ is contained in the cycle space $Z_Y$ of $Y$. In this work, we show how to extend a basis for $Z_X$ to a basis of $Z_Y$.

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 ${\cal C}(G)$.

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