On Degree Properties of Crossing-Critical Families of Graphs
نویسندگان
چکیده
منابع مشابه
On Degree Properties of Crossing-Critical Families of Graphs
Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs which contain arbitrarily often vertices of any prescribed odd degree, for sufficiently large k. From this we derive that, for any set of integers D such that min(D) ≥ 3 and 3, 4 ∈ D, and for all sufficiently large k there exists an infinite k-crossing-critical family such that the numbers in D ar...
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iráň constructed infinite families of k-crossing-critical graphs for every k ≥ 3 and Kochol constructed such families of simple graphs for every k ≥ 2. Richter and Thomassen argued that, for any given k ≥ 1 and r ≥ 6, there are only finitely many simple k-crossingcritical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple k-crossing-critic...
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A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k ≥ 171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree. A graph is k-crossing-cr...
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We show that, for all choices of integers k > 2 and m, there are simple 3connected k-crossing-critical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least 7 in crossing-critical graphs remains open. Furthermore, our newly constructed g...
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A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k, l)-crossing family if there is a pair of edge subsets E1, E2 such that |E1| = k and |E2| = l, the edges in E1 are pairwise crossing, the edges in E2 are pairwise crossing, and every edges in E1 is disjoint to every edge in E2. We conjecture th...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2019
ISSN: 1077-8926
DOI: 10.37236/7753