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## Discussiones Mathematicae Graph Theory

2012 | 32 | 3 | 545-556
Tytuł artykułu

### Light edges in 1-planar graphs with prescribed minimum degree

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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).
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
545-556
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-03-03
poprawiono
2011-09-29
zaakceptowano
2011-09-29
Twórcy
autor
• Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
autor
• Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
Bibliografia
• [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.
Typ dokumentu
Bibliografia
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