On light cycles in plane triangulations

AbstractA subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class T(5) of plane triangulations of minimum degree 5. It is known that each G ϵ T(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C4 and C5 in each G ϵ T(5) follows. We prove here that each G ϵ T(5) contains also light cycles C6, C7, C8 and C9 such that every vertex is of degree at most 11, 17, 29 and 41, respectively. Moreover, we prove that no cycle Ck with k ⩾ 11 is light in the class T(5)