DMGT

Discussiones Mathematicae Graph Theory 32(3) (2012) 545-556
DOI: https://doi.org/10.7151/dmgt.1625

Light Edges in 1-planar Graphs with Prescribed Minimum Degree

Dávid Hudák and Peter Šugerek

Institute of Mathematics, Faculty of Science,
Pavol Jozef Šafárik University,
Jesenná 5, 040 01 Košice, Slovakia

Abstract

A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7,7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each of the considered edge types there are 1-planar graphs whose set of types of edges contains just the selected edge type).

Keywords: light edge, 1-planar graph

2010 Mathematics Subject Classification: 05C10.

References

[1]	O.V. Borodin, Precise lower bound for the number of edges of minor weight in planar maps, Math. Slovaca 42 (1992) 129--142.
[2]	R. Diestel, Graph Theory, Springer, Graduate Texts in Mathematics 173 (2nd ed., Springer-Verlag, New York, 2000).
[3]	I. Fabrici and S. Jendrol', An inequality concerning edges of minor weight in convex 3-polytopes, Discuss. Math. Graph Theory 16 (1996) 81--87, doi: 10.7151/dmgt.1024.
[4]	I. Fabrici and T. Madaras, The structure of 1-planar graphs, Discrete Math. 307 (2007) 854--865, doi: 10.1016/j.disc.2005.11.056.
[5]	D. Hudák and T. Madaras, On local properties of 1-planar graphs with high minimum degree, Ars Math. Contemp. 4 (2011) 245--254.
[6]	D. Hudák and T. Madaras, On local structure of 1-planar graphs of minimum degree 5 and girth 4, Discuss. Math. Graph Theory 29 (2009) 385--400, doi: 10.7151/dmgt.1454.
[7]	J. Ivančo, The weight of a graph, Ann. Discrete Math. 51 (1992) 113--116, doi: 10.1016/S0167-5060(08)70614-9.
[8]	S. Jendrol' and I. Schiermeyer, On max-min problem concerning weights of edges, Combinatorica 21 (2001) 351--359, doi: 10.1007/s004930100001.
[9]	S. Jendrol' and M. Tuhársky, A Kotzig type theorem for non-orientable surfaces, Math. Slovaca 56 (2006) 245--253.
[10]	S. Jendrol', M. Tuhársky and H.-J. Voss, A Kotzig type theorem for large maps on surfaces, Tatra Mt. Math. Publ. 27 (2003) 153--162.
[11]	S. Jendrol' and H.-J. Voss, Light subgraphs of graphs embedded in plane and projective plane --- a survey, Preprint Inst. of Algebra MATH-AL-02-2001, TU Dresden.
[12]	S. Jendrol' and H.-J. Voss, Light subgraph.
[13]	E. Jucovič, Convex polytopes, Veda Bratislava, 1981 (in Slovak).
[14]	A. Kotzig, Contribution to the theory of Eulerian polyhedra, Math. Slovaca 5 (1955) 111--113.
[15]	G. Ringel, Ein Sechsfarbenproblem auf der Kugel, Abh. Math. Sem. Univ. Hamburg 29 (1965) 107--117, doi: 10.1007/BF02996313.
[16]	D.P. Sanders, On light edges and triangles in projective planar graphs, J. Graph Theory 21 (1996) 335--342.

Received 3 March 2011
Revised 29 September 2011
Accepted 29 September 2011