Structure of edges in plane graphs with bounded dual edge weight

Abstract

The weight of an edge e is the degree-sum of its end-vertices. An edge e=uv is an (i,j)-edge if deg(u)≤i and deg(v)≤j. In 1955, Kotzig proved that every 3-connected planar graph contains an edge of weight at most 13. Later, Borodin extended this result to the class of simple planar graphs with minimum degree at least 3. If we consider the class of plane graphs with minimum degree two, the existence of an edge with small weight can be proved under various conditions (for example girth at least five or bounded number of adjacent vertices of degree two). In this paper we investigate the structure of edges in plane graphs with prescribed dual edge weight what is the minimum sum of degrees of two faces sharing an edge. We prove that every plane graph with minimum degree two and dual edge weight at least w⁎ contains an edge of type (2,10) or (3,4) if w⁎=9, (2,10) or (3,3) if w⁎=10, (2,6) or (3,3) if w⁎∈{11,12,13}, (2,6) if w⁎=14, and (2,4) if w⁎≥15. Moreover, all the bounds are the best possible.