Long properly coloured cycles in edge-coloured graphs
نویسندگان
چکیده
منابع مشابه
Properly Coloured Hamiltonian Paths in Edge-coloured Complete Graphs
We consider edge-coloured complete graphs. A path or cycle Q is called properly coloured (PC) if any two adjacent edges of Q differ in colour. Our note is inspired by the following conjecture by B. Bollobás and P. Erdős (1976) : if G is an edge-coloured complete graph on n vertices in which the maximum monochromatic degree of every vertex is less than bn/2c, then G contains a PC Hamiltonian cyc...
متن کاملMulti-Coloured Hamilton Cycles In Random Edge-Coloured Graphs
We de ne a space of random edge-coloured graphs Gn;m; which correspond naturally to edge -colourings of Gn;m. We show that there exist constants K0; K1 21 such that provided m K0n logn and K1n then a random edge coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1, as the number of vertices n tends to in nity.
متن کاملRandom subgraphs of properly edge-coloured complete graphs and long rainbow cycles
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several e...
متن کاملA Note on Alternating Cycles in Edge-Coloured Graphs
Grossman and HH aggkvist gave a characterisation of two-edge-coloured graphs, which have an alternating cycle (i.e. a cycle in which no two consecutive edges have the same colour). We extend their characterisa-tion to edge-coloured graphs, with any number of colours. That is we show that if there is no alternating cycle in an edge-coloured graph, G, then it contains a vertex z, such that there ...
متن کاملPartitioning edge-coloured complete graphs into monochromatic cycles and paths
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...
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ژورنال
عنوان ژورنال: Journal of Graph Theory
سال: 2018
ISSN: 0364-9024
DOI: 10.1002/jgt.22405