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

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.