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Connected subgraphs with small degree sums in 3‐connected planar graphs
Author(s) -
Enomoto Hikoe,
Ota Katsuhiro
Publication year - 1999
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(199903)30:3<191::aid-jgt4>3.0.co;2-x
Subject(s) - combinatorics , mathematics , planar graph , vertex (graph theory) , vertex connectivity , graph , discrete mathematics , degree (music) , physics , acoustics
It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg( x ) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t , every 3‐connected planar graph G with | V ( G )| ≥ t has a connected subgraph H of order t such that Σ x ∈ V ( H ) deg G ( x ) ≤ 8 t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2 t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999