Structure of edges of embedded graphs with minimum degree two

Abstract

AbstractThe weight of an edge is the degree‐sum of its end‐vertices. An edge is an ‐edge if and . In 1955, Kotzig proved that every 3‐connected plane graph contains an edge of weight at most 13. Later, Borodin proved the existence of such an edge in plane graphs with minimum degree at least three. If we consider a graph embedded on a surface with nonpositive Euler characteristic, minimum degree at least three and sufficiently large number of vertices, then the existence of an edge of weight at most 15 can be proved. In this paper we consider graphs with minimum degree two, minimum face degree at least three, embedded on a surface with nonpositive Euler characteristic. We prove that every connected graph with sufficiently large number of vertices, minimum degree two and minimum face degree at least , embedded on a surface with nonpositive Euler characteristic, contains an edge of type , , or if , , or if , or if , if , if , and if . We will also discuss the quality of our results.