منابع مشابه
Monochromatic Cycles in 2-Coloured Graphs
Li, Nikiforov and Schelp [12] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length `, for all ` ∈ [4, dn/2e]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with δ(G) = 3n/4 that do not contain all such cycles. Finally, we show that, for all δ > 0 and n > n0(δ), if G is a 2-e...
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A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertexdisjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r ≥ 3. In contrast to this, we show that in any edge-colouring of a complete g...
متن کاملMonochromatic triangles in three-coloured graphs
In 1959, Goodman [9] determined the minimum number of monochromatic triangles in a complete graph whose edge set is 2-coloured. Goodman [10] also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, for n sufficiently large, we determine the minimum number of monochromatic triangles in a 3-coloured copy of K...
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In 1998 Luczak, Rödl and Szemerédi [7] proved, by means of the Regularity Lemma, that there exists n0 such that, for any n ≥ n0 and two-edge-colouring of Kn, there exists a pair of vertex disjoint monochromatic cycles of opposite colours covering the vertices of Kn. In this paper we make use of an alternative method of finding useful structure in a graph, leading to a proof of the same result w...
متن کاملPartitioning two-coloured complete multipartite graphs into monochromatic paths and cycles
We show that any complete k-partite graph G on n vertices, with k ≥ 3, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of G contains at most n/2 vertices. This extends known results for complete and complete bipartite
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ژورنال
عنوان ژورنال: Combinatorics, Probability and Computing
سال: 2012
ISSN: 0963-5483,1469-2163
DOI: 10.1017/s0963548312000090