The 7-cycle [formula omitted] is light in the family of planar graphs with minimum degree 5

Abstract

A connected graph H is said to be light in the family of graphs H if there exists a positive integer k such that each graph G ∈ H that contains an isomorphic copy of H contains a subgraph K isomorphic to H that satisfies the inequality ∑ v ∈ V ( K ) deg G ( v ) ⩽ k . It is known that an r-cycle C r is light in the family of planar graphs with minimum degree 5 if 3 ⩽ r ⩽ 6 , and not light for r ⩾ 11 . We prove that C 7 is also light in this family.