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- W2767363989 abstract "In 1995, Erdös and Gyárfás proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $pc_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $pc_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours?For fixed integers $r>s$ and as $ntoinfty$, we prove that $pc_{r,s}(K_n) = Theta(n^{1/chi})$, where $chi=max{{1,2+2s-r}}$ is the chromatic number of the Kneser graph $KG(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $alpha$, then we have $pc_{r,s}(G) = O(n^{1/chi})$, where this time around $chi$ is the chromatic number of the Kneser hypergraph $KG^{(alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $alpha$ for which $pc_{r,s}(G) = Omega(n^{1/chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of Sárközy (2012) that $pc_{r,r}(G)$ depends only on $r$ and $alpha$, but not on the number of vertices.We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs." @default.
- W2767363989 created "2017-11-17" @default.
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- W2767363989 date "2018-08-24" @default.
- W2767363989 modified "2023-09-23" @default.
- W2767363989 title "Vertex Covering with Monochromatic Pieces of few Colours" @default.
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- W2767363989 doi "https://doi.org/10.37236/7469" @default.
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