An upper bound on the sum of powers of the degrees of simple 1-planar graphs

Authors:
Július Czap;Jochen Harant;Dávid Hudák
Affiliations:
Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University of Košice, Nmcovej 32, 040 01 Košice, Slovakia;Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany;Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
Venue:
Discrete Applied Mathematics
Year:
2014

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Abstract

A 1-planar graph is a graph that can be drawn in the plane such that each edge is crossed by at most one other edge. For a fixed integer k=2 and a simple 1-planar graph G on n vertices it is proven that 2(n-1)^k+O(n) is an upper bound on the sum of the k-th powers of the degrees of G.