An intrinsic characterization of the direct product of balls

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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On Homology and Cohomology Groups of Remainders

Georgian Mathematical Journal ◽

10.1515/gmj.2004.613 ◽

2004 ◽

Vol 11 (4) ◽

pp. 613-633

Author(s):

V. Baladze ◽

L. Turmanidze

Keyword(s):

Uniform Space ◽

Cohomology Groups ◽

Uniform Spaces ◽

Intrinsic Characterization

Abstract Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.

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On a problem concerning the intrinsic characterization of ? n

Mathematische Annalen ◽

10.1007/bf01352022 ◽

1979 ◽

Vol 246 (1) ◽

pp. 15-22 ◽

Cited By ~ 2

Author(s):

H. Wu

Keyword(s):

Intrinsic Characterization

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Neighborhood Reconstruction and Cancellation of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6676 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

Richard H. Hammack ◽

Cristina Mullican

Keyword(s):

Direct Product ◽

Structural Characterization ◽

Unsolved Problem ◽

Reconstruction Problem ◽

Cancellation Problem ◽

Open Vertex ◽

Isomorphic Graphs ◽

Product Of Graphs

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

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Vertex-Transitive Direct Products of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6999 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Richard H. Hammack ◽

Wilfried Imrich

Keyword(s):

Direct Product ◽

Direct Products ◽

Odd Cycle ◽

Vertex Transitive ◽

Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

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