Some properties and applications of odd-colorable $r$-hypergraphs

Let $r\geq2$ and $r$ be even. An $r$-hypergraph $G$ on $n$ vertices is called odd-colorable if there exists a map $\varphi:[n]\rightarrow\lbrack r]$ such that for any edge $\{j_{1},j_{2},\cdots,j_{r}\}$ of $G$, we have $\varphi(j_{1})+\varphi(j_{2})+\cdot\cdot\cdot+\varphi(j_{r})\equiv r/2(\operatorname{mod}r).$ In this paper, we first determine that, if $r=2^{q}(2t+1)$ and $n\ge 2^{q}(2^{q}-1)r$, then the maximum chromatic number in the class of the odd-colorable $r$-hypergraphs on $n$ vertices is $2^q$, which answers a question raised by V. Nikiforov recently in [V. Nikiforov, Hypergraphs and hypermatrices with symmetric spectrum. Prinprint available in arXiv:1605.00709v2, 10 May, 2016]. We also study some applications of the symmetric spectral property of the odd-colorable $r$-graphs given in that same paper by V. Nikiforov. We show that the Laplacian spectrum and the signless Laplacian spectrum of an $r$-hypergraph $G$ are equal if and only if $G$ is odd-colorable, and then study some further applications of these spectral properties.