Light subgraphs of graphs embedded in the plane and in the projective plane – a survey –
Stanislav Jendrol′, H.‐J. Voss · 2011
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. Recently, Fabrici and Jendrol’ proved that every 3-connected planar graph G that contains a k-vertex path, a path on k vertices, 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 with degree sum ∑ 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 and projective plane graphs and multigraphs. A similar survey dealing with the family of all graphs embedded in surfaces other than the sphere and the projective plane was prepared as well.