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Homeomorphically Irreducible Spanning Trees in Locally Connected Graphs

Published online by Cambridge University Press:  02 February 2012

GUANTAO CHEN ,
HAN REN  and
SONGLING SHAN
GUANTAO CHEN
Affiliation:
Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia 30303, USA (e-mail: [email protected])
HAN REN
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, China
SONGLING SHAN
Affiliation:
Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia 30303, USA (e-mail: [email protected])
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Abstract

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if, for every vertex v ∈ V(G), the subgraph induced by the neighbourhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, we confirm the following conjecture due to Archdeacon: every graph that triangulates some surface has a HIST, which was proposed as a question by Albertson, Berman, Hutchinson and Thomassen.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 107 - 111
Copyright
Copyright © Cambridge University Press 2012

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