scholarly journals Hamilton Circles in Cayley Graphs

10.37236/7009 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Babak Miraftab ◽  
Tim Rühmann
Keyword(s):  
Cayley Graphs ◽  
Infinite Graphs ◽  
Sufficient Condition ◽  
Hamilton Cycles ◽  
Locally Finite

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we prove a sufficient condition for the existence of Hamilton circles in locally finite Cayley 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 Extending Cycles Locally to Hamilton Cycles

10.37236/3960 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Matthias Hamann ◽  
Florian Lehner ◽  
Julian Pott
Keyword(s):  
Unit Circle ◽  
Unit Interval ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Hamilton Cycles ◽  
Free Graph ◽  
Locally Finite ◽  
Hamilton Connected

A Hamilton circle in an infinite graph is a homeomorphic copy of the  unit circle $S^1$ that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are $3$-connected, extending a result of Asratian. Hamilton-connected means that between any two vertices there is a Hamilton arc, a homeomorphic copy of the unit interval $[0,1]$ that contains all vertices and all ends precisely once.

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 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.

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.

Unitary homogeneous bi-Cayley graphs over finite commutative rings

2019 ◽  
Vol 19 (09) ◽  
pp. 2050173
Author(s):  
Xiaogang Liu ◽  
Chengxin Yan
Keyword(s):  
Commutative Ring ◽  
Cayley Graph ◽  
Line Graph ◽  
Cayley Graphs ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Finite Commutative Rings

Let [Formula: see text] denote the unitary homogeneous bi-Cayley graph over a finite commutative ring [Formula: see text]. In this paper, we determine the energy of [Formula: see text] and that of its complement and line graph, and characterize when such graphs are hyperenergetic. We also give a necessary and sufficient condition for [Formula: see text] (respectively, the complement of [Formula: see text], the line graph of [Formula: see text]) to be Ramanujan.

scholarly journals Hamilton Cycles in Planar Locally Finite Graphs

2008 ◽  
Vol 22 (4) ◽  
pp. 1381-1392 ◽  
Author(s):  
Henning Bruhn ◽  
Xingxing Yu
Keyword(s):  
Hamilton Cycles ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs

scholarly journals Some applications of minimal open sets

2001 ◽  
Vol 27 (8) ◽  
pp. 471-476 ◽  
Author(s):  
Fumie Nakaoka ◽  
Nobuyuki Oda
Keyword(s):  
Hausdorff Space ◽  
Nonempty Subset ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Locally Finite ◽  
Open Set ◽  
Finite Space

We characterize minimal open sets in topological spaces. We show that any nonempty subset of a minimal open set is pre-open. As an application of a theory of minimal open sets, we obtain a sufficient condition for a locally finite space to be a pre-Hausdorff space.

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 Integral Cayley Graphs Generated by Distance Sets in Vector Spaces over Finite Fields

10.37236/2730 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Ngoc Dai Nguyen ◽  
Minh Hai Nguyen ◽  
Duy Hieu Do ◽  
Anh Vinh Le
Keyword(s):  
Finite Fields ◽  
Euclidean Distance ◽  
Cayley Graphs ◽  
Vector Spaces ◽  
Sufficient Condition ◽  
Finite Rings ◽  
Distance Sets

Si Li and the fourth listed author (2008) considered unitary graphs attached to the vector spaces over finite rings using an analogue of the Euclidean distance. These graphs are shown to be integral when the cardinality of the ring is odd or the dimension is even. In this paper, we show that the statement also holds for the remaining case: the cardinality of the ring is even and the dimension is odd, by showing a sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields to be integral.

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