| 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 Σx∈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. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999 |
