A note on the weight of triangle in 1-planar graphs with minimum degree 6

Abstract

It is proved that every 1-planar graph with minimum degree at least 6 contains a triangle uvw with d(u)+d(v)+d(w) ≤ 22, or with d(u) = 6, d(v) = 7 and d(w) = 10, or with d(u) = 7,d(v) = 8 and d(w) = 8. Moreover, it is also proved that every plane graph with independent crossings with minimum degree 6 contains a triangle uvw with d(u) = d(v) = d(w) = 6.