### Abstract

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 ∑_{x∈V(H)} deg_{G}(x) ≤ 8t - 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 2t - 1.

Original language | English |
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Pages (from-to) | 191-203 |

Number of pages | 13 |

Journal | Journal of Graph Theory |

Volume | 30 |

Issue number | 3 |

Publication status | Published - 1999 Mar |

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### Keywords

- 3-connected planar graph
- 3-tree
- Connected subgraph
- Degree sum

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*30*(3), 191-203.

**Connected subgraphs with small degree sums in 3-connected planar graphs.** / Enomoto, Hikoe; Ota, Katsuhiro.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 30, no. 3, pp. 191-203.

}

TY - JOUR

T1 - Connected subgraphs with small degree sums in 3-connected planar graphs

AU - Enomoto, Hikoe

AU - Ota, Katsuhiro

PY - 1999/3

Y1 - 1999/3

N2 - 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 ∑x∈V(H) degG(x) ≤ 8t - 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 2t - 1.

AB - 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 ∑x∈V(H) degG(x) ≤ 8t - 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 2t - 1.

KW - 3-connected planar graph

KW - 3-tree

KW - Connected subgraph

KW - Degree sum

UR - http://www.scopus.com/inward/record.url?scp=0033437240&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033437240&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033437240

VL - 30

SP - 191

EP - 203

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 3

ER -