Geodesic
On $n$-stars in colorings and orientations of graphs
Algebra and discrete mathematics, Tome 22 (2016) no. 2, pp. 301-303

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An $n$-star $S$ in a graph $G$ is the union of geodesic intervals $I_{1},\ldots,I_{k}$ with common end $O$ such that the subgraphs $I_{1}\setminus\{O\},\ldots,I_{k}\setminus\{O\}$ are pairwise disjoint and $l(I_{1})+\ldots+l(I_{k})= n$. If the edges of $G$ are oriented, $S$ is directed if each ray $I_{i}$ is directed. For natural number $n,r$, we construct a graph $G$ of $\operatorname{diam}(G)=n$ such that, for any $r$-coloring and orientation of $E(G)$, there exists a directed $n$-star with monochrome rays of pairwise distinct colors.
Keywords: $n$-star, coloring
Mots-clés : orientation. 
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     author = {Igor Protasov},
     title = {On $n$-stars in colorings and orientations of graphs},
     journal = {Algebra and discrete mathematics},
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     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2016_22_2_a9/}
}
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Igor Protasov. On $n$-stars in colorings and orientations of graphs. Algebra and discrete mathematics, Tome 22 (2016) no. 2, pp. 301-303. http://geodesic.mathdoc.fr/item/ADM_2016_22_2_a9/