A conjecture of Gyárfás and Sárközy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $\mathcal{K}_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k-2$ vertices. Recently, the authors affirmed the conjecture. In the note we show that for every $2$-coloring of $\mathcal{K}_n^k$, one can find two monochromatic paths of distinct colors to cover all vertices of $\mathcal{K}_n^k$ such that they share at most $k-2$ vertices. Omidi and Shahsiah conjectured that $R(\mathcal{P}_t^k,\mathcal{P}_t^k) =t(k-1)+\lfloor \frac{t+1}{2}\rfloor$ holds for $k\ge 3$ and they affirmed the conjecture for $k=3$ or $k\ge 8$. We show that if the conjecture is true, then $k-2$ is best possible for our result.
@article{10_37236_6936,
author = {Changhong Lu and Rui Mao and Bing Wang and Ping Zhang},
title = {Cover \(k\)-uniform hypergraphs by monochromatic loose paths},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {4},
doi = {10.37236/6936},
zbl = {1373.05067},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6936/}
}
TY - JOUR
AU - Changhong Lu
AU - Rui Mao
AU - Bing Wang
AU - Ping Zhang
TI - Cover \(k\)-uniform hypergraphs by monochromatic loose paths
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/6936/
DO - 10.37236/6936
ID - 10_37236_6936
ER -
%0 Journal Article
%A Changhong Lu
%A Rui Mao
%A Bing Wang
%A Ping Zhang
%T Cover \(k\)-uniform hypergraphs by monochromatic loose paths
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/6936/
%R 10.37236/6936
%F 10_37236_6936
Changhong Lu; Rui Mao; Bing Wang; Ping Zhang. Cover \(k\)-uniform hypergraphs by monochromatic loose paths. The electronic journal of combinatorics, Tome 24 (2017) no. 4. doi: 10.37236/6936
