Forcing $k$-Repetitions in Degree Sequences
نویسندگان
چکیده
منابع مشابه
Forcing $k$-repetitions in Degree Sequences
One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without 3 vertices of the same degree, it is natural to ask if for any fixed k, every graph G is “close” to a graph G′ with k vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we sho...
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We prove the following approximate version of Pósa’s theorem for directed graphs: every directed graph on n vertices whose inand outdegree sequences satisfy di ≥ i+o(n) and d+i ≥ i+o(n) for all i ≤ n/2 has a Hamilton cycle. In fact, we prove that such digraphs are pancyclic (i.e. contain cycles of lengths 2, . . . , n). We also prove an approximate version of Chvátal’s theorem for digraphs. Thi...
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The existence of unavoidable repeated substructures is a known phenomenon implied by the pigeonhole principle and its generalizations. A fundamental problem is to determine the largest size of a repeated substructure in any combinatorial structure from a given class. The strongest notion of repetition is a pair of isomorphic substructures, such as a pair of vertexdisjoint or edge-disjoint isomo...
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A famous conjecture of Pósa from 1962 asserts that every graph on n vertices and with minimum degree at least 2n/3 contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy and Szemerédi [23]. In this paper we prove a degree sequence version of Pósa’s conjecture: Given any η > 0, every graph G of sufficiently large order n contains the square...
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We consider with a new point of view the notion of nth powers in connection with the k-abelian equivalence of words. For a fixed natural number k, words u and v are k-abelian equivalent if every factor of length at most k occurs in u as many times as in v. The usual abelian equivalence coincides with 1-abelian equivalence. Usually k-abelian squares are defined as words w for which there exist n...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2014
ISSN: 1077-8926
DOI: 10.37236/3503