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Coloring outside the lines: Spectral bounds for generalized hypergraph colorings

Lies Beers, Raffaella Mulas

Abstract

It is known that, for an oriented hypergraph with (vertex) coloring number $χ$ and smallest and largest normalized Laplacian eigenvalues $λ_1$ and $λ_N$, respectively, the inequality $χ\geq (λ_N-λ_1)/\min\{λ_N-1,1-λ_1\}$ holds. We provide necessary conditions for oriented hypergraphs for which this bound is tight. Focusing on $c$-uniform unoriented hypergraphs, we then generalize the bound to the setting of \emph{$d$-proper colorings}: colorings in which no edge contains more than $d$ vertices of the same color. We also adapt our proof techniques to derive analogous spectral bounds for \emph{$d$-improper colorings} of graphs and for edge colorings of hypergraphs. Moreover, for all coloring notions considered, we provide necessary conditions under which the bound is an equality.

Coloring outside the lines: Spectral bounds for generalized hypergraph colorings

Abstract

It is known that, for an oriented hypergraph with (vertex) coloring number and smallest and largest normalized Laplacian eigenvalues and , respectively, the inequality holds. We provide necessary conditions for oriented hypergraphs for which this bound is tight. Focusing on -uniform unoriented hypergraphs, we then generalize the bound to the setting of \emph{-proper colorings}: colorings in which no edge contains more than vertices of the same color. We also adapt our proof techniques to derive analogous spectral bounds for \emph{-improper colorings} of graphs and for edge colorings of hypergraphs. Moreover, for all coloring notions considered, we provide necessary conditions under which the bound is an equality.

Paper Structure

This paper contains 12 sections, 21 theorems, 130 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The normalized Laplacian for hypergraphs
  4. Coloring hypergraphs
  5. Eigenvalue bounds for strong hypergraph vertex coloring
  6. Strong vertex coloring of general oriented hypergraphs
  7. Strong vertex coloring of unoriented c-uniform hypergraphs
  8. Spectral bounds for a generalized notion of hypergraph coloring
  9. d-proper colorings of c-uniform unoriented hypergraphs
  10. q-tailored colorings
  11. Coloring graphs d-improperly
  12. Edge-coloring hypergraphs

Key Result

Theorem 2.16

Let $H$ be an $N$-dimensional vector space with a positive definite scalar product $(\cdot,\cdot)$, and let $A\colon H\rightarrow H$ be a self-adjoint linear operator. Let $\mathcal{H}_{k}$ be the family of all $k$-dimensional subspaces of $H$. Then the eigenvalues $\lambda_{1}\leq \ldots \leq \lamb The vectors $g_k$ realizing such a min-max or max-min then are corresponding eigenvectors, and the

Figures (4)

  • Figure 1: Two examples of hypergraphs: a disconnected hypergraph with five disjoint edges (left) and a hyperflower with five petals (right).
  • Figure 2: An oriented hypergraph with $A=\boldsymbol 0$.
  • Figure 3: The two $3$-uniform hyperflowers $H_{3,1}^3$ (left) and $H_{3,2}^3$ (right) with $3$ petals and with $1$ and $2$ central vertices, respectively.
  • Figure 4: The $9$-uniform hyperflower $H_{3,2}^9$, for which the bound in \ref{['thm:d-proper']} is sharp.

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 57 more