Light subgraphs of graphs embedded in the plane—A survey

Abstract

It is well known that every planar graph contains a vertex of degree at most 5. A theorem of Kotzig states that every 3-connected planar graph contains an edge whose endvertices have degree-sum at most 13. Fabrici and Jendrol’ proved that every 3-connected planar graph G that contains a k-vertex path contains also a k-vertex path P such that every vertex of P has degree at most 5k. A result by Enomoto and Ota says that every 3-connected planar graph G of order at least k contains a connected subgraph H of order k such that the degree sum of vertices of H in G is at most 8k−1. Motivated by these results, a concept of light graphs has been introduced. A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there is an integer w(H,G) such that each member G of G with a copy of H also has a copy K of H such that ∑v∈V(K)degG(v)≤w(H,G).In this paper we present a survey of results on light graphs in different families of plane graphs and multigraphs. A similar survey dealing with the family of all graphs embedded in surfaces other than the sphere was prepared as well.