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DP color functions of hypergraphs

Ruiyi Cui, Liangxia Wan, Fengming Dong

TL;DR

This work extends DP-coloring to hypergraphs by defining the DP color function $P_{DP}({\mathcal{H}},k)$ as the minimum number of colorings over all $k$-fold covers, and it establishes a general upper bound for connected $r$-uniform hypergraphs. It proves that the DP color function equals the usual chromatic counting bound if and only if the hypergraph is a hypertree, and it gives an explicit formula for the DP color function of unicycle linear $r$-uniform hypergraphs, with an odd/even cycle dichotomy; the $r=2$ case recovers known graph results. The results connect DP-coloring behavior in hypergraphs to classical chromatic polynomials and set the stage for further exploration of equality conditions and asymptotics in hypergraph DP-coloring. These insights advance the theory of DP-coloring beyond graphs and suggest several open problems on characterization and thresholds for equality with the chromatic polynomial.

Abstract

In this article, we introduce the DP color function of a hypergraph, based on the DP coloring introduced by Bernshteyn and Kostochka, which is the minimum value where the minimum is taken over all its k-fold covers. It is an extension of its chromatic polynomial. we obtain an upper bound for the DP color functions of hypergraphs when hypergraphs are connected r-uniform hypergraphs for any r greater than one. The upper bound is attained if and only if the hypergraph is a r-uniform hypertree. We also show the cases of the DP color function equal to its chromatic polynomial. These conclusions coincide with the known results of graphs.

DP color functions of hypergraphs

TL;DR

This work extends DP-coloring to hypergraphs by defining the DP color function as the minimum number of colorings over all -fold covers, and it establishes a general upper bound for connected -uniform hypergraphs. It proves that the DP color function equals the usual chromatic counting bound if and only if the hypergraph is a hypertree, and it gives an explicit formula for the DP color function of unicycle linear -uniform hypergraphs, with an odd/even cycle dichotomy; the case recovers known graph results. The results connect DP-coloring behavior in hypergraphs to classical chromatic polynomials and set the stage for further exploration of equality conditions and asymptotics in hypergraph DP-coloring. These insights advance the theory of DP-coloring beyond graphs and suggest several open problems on characterization and thresholds for equality with the chromatic polynomial.

Abstract

In this article, we introduce the DP color function of a hypergraph, based on the DP coloring introduced by Bernshteyn and Kostochka, which is the minimum value where the minimum is taken over all its k-fold covers. It is an extension of its chromatic polynomial. we obtain an upper bound for the DP color functions of hypergraphs when hypergraphs are connected r-uniform hypergraphs for any r greater than one. The upper bound is attained if and only if the hypergraph is a r-uniform hypertree. We also show the cases of the DP color function equal to its chromatic polynomial. These conclusions coincide with the known results of graphs.

Paper Structure

This paper contains 7 sections, 12 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chromatic polynomials of some hypergraphs
  4. DP-colorings of a hypergraph
  5. Some properties of the DP color function
  6. DP color functions of hypertrees and unicycle hypergraphs
  7. Further study

Key Result

Lemma 1

If $\mathcal{T}_{m}^{r}$ is a $r$-uniform hypertree with $m$ edges, where $r\geq 2$ and $m\geq 0$, then

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Proposition 2
  • ...and 6 more