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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2012 Volume 18, Number 3, Pages 26–29 (Mi timm835)

This article is cited in 1 paper

A note on the extendability of an isomorphism of subgraphs of a graph to an automorphism of the graph

V. I. Trofimovab

a Institute of Mathematics and Mechanics, UB Russian Academy of Sciences
b Institute of Mathematics and Computer Sciences, Ural Federal University

Abstract: Let $\Gamma$ be an undirected connected locally finite graph such that its automorphism group is vertex-transitive and has finite vertex stabilizers. For a vertex $v$ of $\Gamma$ and a non-negative integer $n$, let $\langle B_\Gamma(v,n)\rangle_\Gamma$ denote the subgraph of $\Gamma$ generated by the ball $B_\Gamma(v,n)$ of radius $n$ with center $v$. We prove that there exists a non-negative integer $c$ (depending only on $\Gamma$) such that, for any vertices $x$ and $y$ of $\Gamma$ and any non-negative integer $r$, if an isomorphism of $\langle B_\Gamma(x,r)\rangle_\Gamma$ onto $\langle B_\Gamma(y,r)\rangle_\Gamma$ can be extended to an isomorphism of $\langle B_\Gamma(x,r+c)\rangle_\Gamma$ onto $\langle B_\Gamma(y,r+c)\rangle_\Gamma$, then it can also be extended to an automorphism of $\Gamma$. Furthermore, we give a “formula” for $c$. In such a form the result can also be of interest for finite graphs $\Gamma$.

Keywords: vertex-symmetric graph, extension of automorphism.

UDC: 512.54+519.17

Received: 20.01.2012

Language: English



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