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

Authors:
S. Jendrol;H.-J. Voss
Affiliations:
Department of Geometry and Algebra, P. J. Šafárik University and Institute of Mathematics, Slovak Academy of Sciences, Jesenná 5, 041 54 Košice, Slovakia;Department of Algebra, Technical University Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany
Venue:
European Journal of Combinatorics - Special issue: Topological graph theory II
Year:
2005

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Abstract

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) If G has more than 83|χ(M)| vertices then G contains a 3-cycle of weight at most 18. (ii) If Σdeg(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.