منابع مشابه
Describing 3-paths in normal plane maps
We prove that every normal plane map, as well as every 3polytope, has a path on three vertices whose degrees are bounded from above by one of the following triplets: $(3,3,\infty)$, $(3,4,11)$, $(3,7,6)$, $(3,10,4)$, $(3,15,3)$, $(4,4,9)$, $(6,4,8)$, $(7,4,7)$, and $(6,5,6)$. No parameter of this description can be improved, as shown by appropriate 3-polytopes. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
متن کامل5-stars of Low Weight in Normal Plane Maps with Minimum Degree 5
It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48.
متن کاملShort cycles of low weight in normal plane maps with minimum degree 5
In this note, precise upper bounds are determined for the minimum degree-sum of the vertices of a 4-cycle and a 5-cycle in a plane triangulation with minimum degree 5: w(C4) ≤ 25 and w(C5) ≤ 30. These hold because a normal plane map with minimum degree 5 must contain a 4-star with w(K1,4) ≤ 30. These results answer a question posed by Kotzig in 1979 and recent questions of Jendrol’ and Madaras.
متن کاملBraced edges in plane triangulations
A plane triangulation is an embedding of a maximal planar graph in the Euclidean plane. Foulds and Robinson (1979) first studied the problem of transforming one triangulation to another by a sequence of diagonal operations. where a diagonal operation deletes one edge and inserts the other diagonal of the resulting quadrilateral face. An edge which cannot be removed by a single diagonal operatio...
متن کاملOn the structural result on normal plane maps
We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree ∆(G) ≤ D, D ≥ 12 is proved to be upper bounded by 6+ 2D+12 D−2 ((D−1)(t−1)−1). This improves a recent bound 6 + 3D+3 D−2 ((D − 1)t−1 − 1), D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2016
ISSN: 0012-365X
DOI: 10.1016/j.disc.2015.12.021