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
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U2 - 10.1002/(SICI)1097-0118(199903)30:3<191::AID-JGT4>3.0.CO;2-X
DO - 10.1002/(SICI)1097-0118(199903)30:3<191::AID-JGT4>3.0.CO;2-X
M3 - Article
AN - SCOPUS:0033437240
SN - 0364-9024
VL - 30
SP - 191
EP - 203
JO - Journal of Graph Theory
JF - Journal of Graph Theory
IS - 3
ER -