On vertex-degree restricted subgraphs in polyhedral graphs
Discrete Mathematics
Subgraphs of graphs on surfaces with high representativity
Journal of Combinatorial Theory Series B
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It is well-known that every planar graph has a vertex of degreeat most five. Kotzig proved that every 3-connected planar graph hasan edge xy such that deg(x) + deg (y) ≤ 13. Inthis article, considering a similar problem for the case of threeor more vertices that induce a connected subgraph, we show that,for a given positive integer t, every 3-connected planargraph G with |V(G)| ≥ t has aconnected subgraph H of order t such thatΣxεV(H)degG(x) ≤ 8t - 1. As a tool forproving this result, we consider decompositions of 3-connectedplanar graphs into connected subgraphs of order at least tand at most 2t - 1. © 1999 John Wiley & Sons, Inc.J Graph Theory 30: 191203, 1999