Light subgraphs of order at most 3 in large maps of minimum degree 5 on compact 2-manifolds

We investigate the existence of subgraphs H of low degree sum wG(H) of their vertices in graphs G of minimum degree 5 on compact 2-manifolds M of Euler characteristic χ(M)≤0. The value wG(H) is said to be the weight of H in G. We prove:(i) as more than 83|χ(M)| vertices then G contains a 3-cycle of weight at most 18. g(v)≥7(degG(v)−6)>24|χ(M)| then G contains a 3-cycle of weight at most 18 and a path with three vertices of weight at most 17.