1,812 research outputs found
Rainbow Hamilton cycles in random regular graphs
A rainbow subgraph of an edge-coloured graph has all edges of distinct
colours. A random d-regular graph with d even, and having edges coloured
randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with
probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page
Rainbow Matchings and Hamilton Cycles in Random Graphs
Let be drawn uniformly from all -uniform, -partite
hypergraphs where each part of the partition is a disjoint copy of . We
let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge
randomly from one of \k colors. We show that if \k=n and where
is sufficiently large then w.h.p. there is a rainbow colored perfect
matching. I.e. a perfect matching in which every edge has a different color. We
also show that if is even and where is sufficiently large
then w.h.p. there is a rainbow colored Hamilton cycle in . Here
denotes a random edge coloring of with colors.
When is odd, our proof requires m=\om(n\log n) for there to be a rainbow
Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
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