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<p>An <i>r</i>-uniform hypergraph is linear if every two edges intersect in at most one vertex. The <i>r</i>- expansion <i>F</i><sup><i>r</i></sup> of a graph <i>F</i> is the <i>r</i>-uniform hypergraph obtained from <i>F</i> by enlarging each edge of <i>F</i> with a vertex subset of size <i>r</i> - 2 disjoint from the vertex set of <i>F</i> such that distinct edges are enlarged by disjoint subsets. Let ex<Stack> <sub><i>r</i></sub> <sup>lin</sup> </Stack>(<i>n</i>; <i>F</i><sup><i>r</i></sup>) and spex<Stack> <sub><i>r</i></sub> <sup>lin</sup> </Stack>(<i>n</i>; <i>F</i><sup><i>r</i></sup>) be the maximum number of edges and the maximum spectral radius of all <i>F</i><sup><i>r</i></sup>-free linear <i>r</i>-uniform hypergraphs with <i>n</i> vertices, respectively. In this paper, we present sharp (or asymptotic) bounds of ex<Stack> <sub><i>r</i></sub> <sup>lin</sup> </Stack>(<i>n</i>; <i>F</i><sup><i>r</i></sup>) and spex<Stack> <sub><i>r</i></sub> <sup>lin</sup> </Stack>(<i>n</i>; <i>F</i><sup><i>r</i></sup>) by establishing a connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where <i>F</i> is a (<i>k</i> + 1)-color critical graph or a graph with chromatic number <i>k</i>.</p>

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Linear Spectral Turán Problems for Expansions of Graphs with Given Chromatic Number

  • 期刊论文

  • 发表日期：

    2025年5月30日

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  • Chuan-ming She,
  • Yi-zheng Fan,
  • Liying Kang,
  • Yaoping Hou

摘要

An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. The r- expansion Fr of a graph F is the r-uniform hypergraph obtained from F by enlarging each edge of F with a vertex subset of size r - 2 disjoint from the vertex set of F such that distinct edges are enlarged by disjoint subsets. Let ex r lin (n; Fr) and spex r lin (n; Fr) be the maximum number of edges and the maximum spectral radius of all Fr-free linear r-uniform hypergraphs with n vertices, respectively. In this paper, we present sharp (or asymptotic) bounds of ex r lin (n; Fr) and spex r lin (n; Fr) by establishing a connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where F is a (k + 1)-color critical graph or a graph with chromatic number k.