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• W585142893 abstract "In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{v{r}}{'{i}}{v{z}} bound. Next, we introduce multiple Kneser hypergraphs and we specify the chromatic number of some multiple Kneser hypergraphs. For a vector of positive integers $vec{s}=(s_1,s_2,ldots,s_m)$ and a partition $pi=(P_1,P_2,ldots,P_m)$ of ${1,2,ldots,n}$, the multiple Kneser hypergraph ${rm KG}^r(pi; vec{s};k)$ is a hypergraph with the vertex set $$V=left{A: Asubseteq P_1cup P_2cupcdots cup P_m, |A|=k, forall 1leq ileq m; |Acap P_i|leq s_iright}$$ whose edge set is consist of any $r$ pairwise disjoint vertices. We determine the chromatic number of multiple Kneser hypergraphs provided that $r=2$ or for any $1leq ileq m$, we have $|P_i|leq 2s_i$. A subset $S subseteq [n]$ is almost $s$-stable if for any two distinct elements $i,jin S$, we have $|i-j|geq s$. The almost $s$-stable Kneser hypergraph ${rm KG}^r(n,k)_{s-stab}^{sim}$ has all $s$-stable subsets of $[n]$ as the vertex set and every $r$-tuple of pairwise disjoint vertices forms an edge. Meunier [The chromatic number of almost stable Kneser hypergraphs. J. Combin. Theory Ser. A, 118(6):1820--1828, 2011] showed for any positive integer $r$, $chi({rm KG}^r(n,k)_{2-stab}^{sim})=leftlceil {n-r(k-1) over r-1}rightrceil$. We extend this result to a large family of Schrijver hypergraphs. Finally, we present a colorful-type result which confirms the existence of a completely multicolored complete bipartite graph in any coloring of a graph." @default.
• W585142893 created "2016-06-24" @default.
• W585142893 creator A5053121292 @default.
• W585142893 creator A5070993532 @default.
• W585142893 date "2015-11-01" @default.
• W585142893 modified "2023-10-14" @default.
• W585142893 title "On the chromatic number of general Kneser hypergraphs" @default.
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• W585142893 doi "https://doi.org/10.1016/j.jctb.2015.05.010" @default.
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