Zero-free intervals of chromatic polynomials of mixed hypergraphs

TL;DR

This paper establishes new zero-free intervals for chromatic polynomials of certain hypergraph families, extending known results from graphs to hypergraphs and identifying specific intervals where these polynomials do not have zeros.

Contribution

It proves zero-free intervals for chromatic polynomials of hypergraphs, generalizing previous graph results and identifying specific intervals for hypergraph families.

Findings

(-∞, 0) is zero-free for hypergraph chromatic polynomials

(0, 1) is zero-free for a subfamily of hypergraphs

Extends known graph zero-free results to hypergraphs

Abstract

In this paper, we prove that $(-\infty, 0)$ is a zero-free interval for chromatic polynomials of a family ${\cal L}_0$ of hypergraphs and $(0, 1)$ is a zero-free interval for chromatic polynomials of a subfamily ${\cal L}_0'$ of ${\cal L}_0$ of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

Figures

Equations

$$P(\mathcal{H},\lambda )={\lambda }^6(\lambda -1{)}^4({\lambda }^3-2)({\lambda }^2+\lambda +1{)}^2,$$

$$P(\mathcal{H},\lambda )=\lambda (\lambda -1)(\lambda -2)(\lambda (\lambda -2{)}^2-1),$$

$$P(\mathcal{H},\lambda )=P(\mathcal{H}-e_2,\lambda ),$$

$$P(\mathcal{H},\lambda )=P({\mathcal{H}}^{\ast },\lambda ).$$

$$\{e\in \mathcal{E}:e\cap V_0=\varnothing \}\cup \{(e-V_0)\cup \{w\}:e\in \mathcal{E},e\cap V_0\ne \varnothing \},$$

$$P(\mathcal{H},\lambda )=P(\mathcal{H}-e,\lambda )-P(\mathcal{H}/e,\lambda ).$$

$$(\mathcal{E}/e_0{)}^{\ast }\subseteq \{e\in \mathcal{E}:e\cap e_0=\varnothing \}\cup \{(e-e_0)\cup \{w\}:|e\cap e_0|=1,e\in \mathcal{E}\},$$

$$\mathcal{E}/e_0=\{e\in \mathcal{E}:e\cap e_0=\varnothing \}\cup \{(e-e_0)\cup \{w\}:e\in \mathcal{E},e\cap e_0\ne \varnothing \}.$$

$$P(\mathcal{H},\lambda )=P(\mathcal{H}-e_0,\lambda )-P(\mathcal{H}/e_0,\lambda )=P(\mathcal{H}-e_0,\lambda )-P((\mathcal{H}/e_0{)}^{\ast },\lambda ).$$

$$(-1{)}^{n}P(\mathcal{H}-e_0,\lambda )>0\text{ and }(-1{)}^{n-|e_0|+1}P((\mathcal{H}/e_0{)}^{\ast },\lambda )>0.$$

$$(-1{)}^{n}P(\mathcal{H},\lambda )=$$

$$=$$

$$>$$

$$Q(\mathcal{H},\lambda )=\frac{1}{\lambda }\cdot P(\mathcal{H},\lambda ).$$

$$(-1{)}^{|V|+1}Q(\mathcal{H},\lambda )>0.$$

$$P(\mathcal{H},\lambda )=P(\mathcal{H}-e_0,\lambda )-P(\mathcal{H}/e_0,\lambda )=P(\mathcal{H}-e_0,\lambda )-P((\mathcal{H}/e_0{)}^{\ast },\lambda ).$$

$$Q(\mathcal{H},\lambda )=Q(\mathcal{H}-e_0,\lambda )-Q((\mathcal{H}/e_0{)}^{\ast },\lambda ).$$

$$(-1{)}^{|V|+1}Q(\mathcal{H}-e_0,\lambda )>0\text{ and }(-1{)}^{|V|-|e_0|+2}Q((\mathcal{H}/e_0{)}^{\ast },\lambda )>0.$$

$$(-1{)}^{|V|+1}Q(\mathcal{H},\lambda )=$$

$$=$$

$$>$$

Full text

MSC: 05C15, 05C31, 05C65

Keywords: graph; hypergraph; chromatic polynomial

1 Introduction

For any simple graph $G=(V,E)$ and any positive integer $k$, a proper $k$-coloring of $G$ is a mapping $\phi:V\rightarrow\{1,2,\ldots,k\}$ such that $\phi(u)\neq\phi(v)$ holds for each pair of adjacent vertices $u$ and $v$ in $G$. The chromatic polynomial $P(G,\lambda)$ of $G$ is the function which counts the number of proper $\lambda$-colorings of $G$ whenever $\lambda$ is a positive integer. It was introduced by Birkhoff [6] in 1912 for planar graphs, in hope of proving the four-color conjecture, and extended to all graphs by Whitney [36] in 1932. Although Birkhoff’s attempt failed, the study of chromatic polynomials is one of the active research areas in graph theory, especially the study of the zero distribution of chromatic polynomials  [13, 14, 22, 24, 37, 38].

It was proved by Sokal [23] in 2004 that the zeros of chromatic polynomials of graphs are dense in the whole complex plane. It is also known that $(-\infty,0),(0,1)$ and $(1,32/27]$ are the only three maximal zero-free real intervals for chromatic polynomials of graphs (see [13, 24, 31] ), where the zero-free interval $(1,32/27]$ was found by Jackson [13] and was proved to be the last such interval by Thomassen [24]. For certain subsets of graphs, Thomassen [25] showed that if $G$ is any graph with a Hamiltonian path, then $P(G,\lambda)$ is zero-free in the interval $(1,1.2955\cdots)$, while Dong and Jackson [9] proved that for any 3-connected plannar graph $G$, $P(G,\lambda)$ is zero-free in the interval $(1,1.2040\cdots)$.

The study of chromatic polynomials has been extended to chromatic polynomials of hypergraphs for many years. A hypergraph $\mathcal{H}$ is an ordered pair $(V(\mathcal{H}),\mathcal{E}(\mathcal{H}))$, where $V(\mathcal{H})$ is a finite set, called the vertex set of $\mathcal{H}$, and $\mathcal{E}(\mathcal{H})$ is a set of non-empty subsets of $V(\mathcal{H})$, called the edge set of $\mathcal{H}$. Obviously, $\mathcal{E}(\mathcal{H})\subseteq\{e\subseteq V(\mathcal{H}):|e|\geq 1\}$. Thus a graph is a hypergraph $\mathcal{H}$ with $|e|\leq 2$ for each $e\in\mathcal{E}(\mathcal{H})$. Regarding colorings of hypergraphs, there are several different definitions, such as strong proper colorings [1] and $\mathcal{C}$-colorings [32]. In this paper, we work with weak proper colorings of hypergraphs which were introduced by Erdős and Hajnal [11] in 1966. For any integer $k\geq 1$, a weak proper $k$-coloring of a hypergraph $\mathcal{H}=(V,\mathcal{E})$ is an assignment of colors from $\{1,\ldots,k\}$ to the vertices so that each edge contains at least two vertices with different colors, i.e., there is a mapping $\phi:V\rightarrow\{1,\ldots,k\}$ such that $|\{\phi(v):v\in e\}|\geq 2$ holds for each $e\in\mathcal{E}$. The chromatic polynomial of $\mathcal{H}$, denoted by $P(\mathcal{H},\lambda)$, is the function which counts the number of weak proper $k$-colorings of $\mathcal{H}$ whenever $\lambda=k$ is a positive integer, and it is indeed a polynomial in $\lambda$ [8, 26].

A weak proper $k$-coloring of a hypergraph $\mathcal{H}$ is actually a proper $k$-coloring of a graph whenever $\mathcal{H}$ is a graph. In such case, $P(\mathcal{H},\lambda)$ is a chromatic polynomial of a graph. Thus chromatic polynomials of hypergraphs are generalizations of chromatic polynomials of graphs. Many properties of chromatic polynomials of graphs have been extended to chromatic polynomials of hypergraphs (for example, see [2, 3, 4, 7, 8, 26, 27, 28, 30, 29, 33, 34, 35]). Meanwhile in [39], the authors of this article also showed that some properties of chromatic polynomials of hypergraphs do not hold for chromatic polynomials of graphs. One of these properties is that chromatic polynomials of hypergraphs have real zeros that are dense in the whole set of real numbers, while chromatic polynomials of graphs have three zero-free intervals. It is natural for people to look for a suitable way of extending the result that $(-\infty,0)$ and $(0,1)$ are zero-free intervals of chromatic polynomials of graphs to chromatic polynomials of hypergraphs.

A cycle $C$ in $\mathcal{H}$ is defined to be a sequence of alternating vertices and edges: $(v_{1},e_{1},v_{2},e_{2},\ldots,v_{t},e_{t},v_{1}),$ where $t\geq 2$, $v_{1},\ldots,v_{t}$ are pairwise distinct vertices and $e_{1},\ldots,e_{t}$ are pairwise distinct edges, such that $\{v_{i},v_{i+1}\}\subseteq e_{i}$ for $i\in\{1,\ldots,t\}$, where $v_{t+1}=v_{1}$. Sometimes, a cycle $C$ is also written as $(e_{1},\ldots,e_{t})$ for short. In this case, we assume that there exist pairwise distinct vertices $v_{1},\ldots,v_{t}$ such that $v_{i+1}\in e_{i}\cap e_{i+1}$ for all $i=0,1,\ldots,t-1$, where $e_{0}=e_{t}$.

Let $\mathcal{L}_{1}$ be the set of hypergraphs in which each edge has an even size and each cycle $C=(e_{1},\ldots,e_{r})$ contains $e_{i}$ with $|e_{i}|=2$ for some $i\in\{1,\ldots,r\}$. Dohmen [8] proved that for any hypergraph $\mathcal{H}$ in $\mathcal{L}_{1}$, $P(\mathcal{H},\lambda)\neq 0$ holds for any real $\lambda\in(-\infty,0)$.

Theorem 1.1** ([8])**

If $\mathcal{H}=(V,\mathcal{E})$ is a hypergraph in $\mathcal{L}_{1}$, then $(-1)^{|V|}P(\mathcal{H},\lambda)>0$ holds for all real $\lambda\in(-\infty,0)$.

Theorem 1.1 certainly includes the fact that chromatic polynomials of graphs has no real zero in the interval $(-\infty,0)$. As we know, this result is currently the only known result on zero-free intervals for chromatic polynomials of hypergraphs.

Observe that the hypergraph $\mathcal{H}$ in Figure 1 does not belong to $\mathcal{L}_{1}$, as there is a cycle $(e_{1},e_{2},e_{3})$ in $\mathcal{H}$ with $|e_{i}|>2$ for all $i=1,2,3$. It can be calculated that

[TABLE]

which shows that although $\mathcal{H}$ is not in $\mathcal{L}_{1}$, its chromatic polynomial $P(\mathcal{H},\lambda)$ is zero-free in the interval $(-\infty,0)$.

In this paper, we first extend Dohmen’s result to a larger set of hypergraphs, which definitely includes the hypergraph shown in Figure 1. Futher we prove the existence of an infinite set of hypergraphs with at least one edge of size greater than $2$ whose chromatic polynomials are zero-free in the interval $(0,1)$.

Let $\mathcal{L}_{0}$ be the set of hypergraphs $\mathcal{H}$ in which each edge has an even size and each cycle $C=(e_{1},\ldots,e_{r})$ contains distinct vertices $u$ and $v$ in the set $\bigcup_{1\leq i\leq r}e_{i}$ forming an edge in $\mathcal{H}$. For instance, the hypergraph $\mathcal{H}$ shown in Figure 1 is in $\mathcal{L}_{0}\setminus\mathcal{L}_{1}$, which also implies that $\mathcal{L}_{1}$ is a proper subset of $\mathcal{L}_{0}$. Actually there are infinitely many hypergraphs in $\mathcal{L}_{0}\setminus\mathcal{L}_{1}$. Let $\mathcal{L}_{0}^{\prime}$ be the set of hypergraphs $\mathcal{H}=(V,\mathcal{E})\in\mathcal{L}_{0}$ containing a connected spanning subhypergraph $(V,\mathcal{E}_{2})$, where $\mathcal{E}_{2}=\{e\in\mathcal{E}:|e|=2\}$. Clearly, $\mathcal{L}_{0}^{\prime}\subset\mathcal{L}_{0}$.

We prove the following two results on the chromatic polynomials of hypergraphs in $\mathcal{L}_{0}$ and $\mathcal{L}^{\prime}_{0}$ in this article.

Theorem 1.2

If $\mathcal{H}=(V,\mathcal{E})$ is a hypergraph in $\mathcal{L}_{0}$, then $(-1)^{|V|}P(\mathcal{H},\lambda)>0$ holds for all real $\lambda\in(-\infty,0)$.

Theorem 1.2 implies Theorem 1.1 directly, as $\mathcal{L}_{1}\subset\mathcal{L}_{0}$.

Theorem 1.3

If $\mathcal{H}=(V,\mathcal{E})$ is a hypergraph in $\mathcal{L}_{0}^{\prime}$, then

1. (i)

$(-1)^{|V|+1}P(\mathcal{H},\lambda)>0$* holds for all real $\lambda\in(0,1)$; and* 2. (ii)

$P(\mathcal{H},\lambda)$* has no multiple zero at $\lambda=0$.*

Note that Theorem 1.3 fails for some hypergraphs in $\mathcal{L}_{0}\setminus\mathcal{L}_{0}^{\prime}$. For instance, the hypergraph $\mathcal{H}$ shown in Figure 2 belongs to $\mathcal{L}_{0}\setminus\mathcal{L}_{0}^{\prime}$ and its chromatic polynomial is

[TABLE]

which has a real zero around 0.38, implying that Theorem 1.3 fails for this hypergraph.

The proofs of Theorems 1.2 and 1.3 are given in Section 4.

2 Preliminaries

In this section, we present several known results on chromatic polynomials of hypergraphs, which will be applied later. The first one follows directly from the definition of weak proper colorings of a hypergraph.

Proposition 2.1

[17]* Let $e_{1}$ and $e_{2}$ be two edges in a hypergraph $\mathcal{H}$. If $e_{1}\subseteq e_{2}$, then*

[TABLE]

where $\mathcal{H}-e_{2}$ is the hypergraph obtained from $\mathcal{H}$ by removing $e_{2}$.

Proposition 2.1 shows that we need only to consider Sperner hypergraphs when studying chromatic polynomials of hypergraphs, where a hypergraph $\mathcal{H}=(V,\mathcal{E})$ is called Sperner if $e_{1}\nsubseteq e_{2}$ holds for any distinct edges $e_{1},e_{2}\in\mathcal{E}$.

For any hypergraph $\mathcal{H}=(V,\mathcal{E})$, its Sperner subhypergraph is defined to be $\mathcal{H}^{*}=(V,\mathcal{E}^{*})$, where $\mathcal{E}^{*}$ is the minimal subset of $\mathcal{E}$ such that for each $e\in\mathcal{E}\setminus\mathcal{E}^{*}$, there exists $e^{\prime}\in\mathcal{E}^{*}$ with $e^{\prime}\subseteq e$. That means, $\mathcal{H}^{*}$ can be obtained from $\mathcal{H}$ by removing any edge $e\in\mathcal{E}$ whenever $e^{\prime}\subseteq e$ holds for another edge $e^{\prime}\in\mathcal{E}$. It is not difficult to show that $\mathcal{E}^{*}$ is uniquely determined by $\mathcal{E}$. Obviously, $\mathcal{H}^{*}$ is Sperner, and $\mathcal{H}^{*}=\mathcal{H}$ when $\mathcal{H}$ is Sperner. Moreover, by Proposition 2.1,

[TABLE]

The following proposition can be obtained from the definition of Sperner hypergaphs directly.

Proposition 2.2

If $\mathcal{H}=(V,\mathcal{E})$ is Sperner and $e_{0}$ is an edge in $\mathcal{H}$ with $|e_{0}|=2$, then $|e_{0}\cap e|\leq 1$ holds for any edge $e\in\mathcal{E}\setminus\{e_{0}\}$.

Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph and $V_{0}\subset V$. Let $\mathcal{H}\cdot V_{0}$ denote the hypergraph obtained from $\mathcal{H}$ by identifying all vertices in $V_{0}$ as one, i.e., $\mathcal{H}\cdot V_{0}$ is the hypergraph with vertex set $(V-V_{0})\cup\{w\}$ and edge set

[TABLE]

where $w$ is the new vertex produced when $\mathcal{H}\cdot V_{0}$ is obtained from $\mathcal{H}$ by identifying all vertices in $V_{0}$. For an edge $e$ in $\mathcal{H}$, let $\mathcal{H}/e$ denote the hypergraph $(\mathcal{H}-e)\cdot e$. We also say that $\mathcal{H}/e$ is obtained from $\mathcal{H}$ by contracting the edge $e$.

The deletion-contraction formula for chromatic polynomials of graphs is very important for the computation of this polynomial [5, 6, 10, 20, 21]. It was extended to chromatic polynomials of hypergraphs by Jones [16].

Theorem 2.3

[16]* Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph. For any $e\in\mathcal{E}$,*

[TABLE]

3 $\mathcal{L}_{0}$ is deletion and contraction closed

In this section, we show that for any Sperner hypergraph $\mathcal{H}\in\mathcal{L}_{0}$ with at least one edge, there is an edge $e_{0}$ in $\mathcal{H}$ such that both $\mathcal{H}-e_{0}$ and $(\mathcal{H}/e_{0})^{*}$ belong to $\mathcal{L}_{0}$.

For any hypergraph $\mathcal{H}=(V,\mathcal{E})$ and $e\in\mathcal{E}$, let $\mathcal{H}/e$ be written as $(V/e,\mathcal{E}/e)$. By the notation for the Sperner subhypergraph, $(\mathcal{H}/e)^{*}$ is written as $(V/e,(\mathcal{E}/e)^{*})$.

Lemma 3.1

Let $\mathcal{H}=(V,\mathcal{E})\in\mathcal{L}_{0}$ and $e_{0}\in\mathcal{E}$. Assume that $\mathcal{H}$ is Sperner. Then,

[TABLE]

where $w\notin V$.

Proof.  By the definition of $\mathcal{H}/e_{0}$, we have

[TABLE]

By the definition of $(\mathcal{H}/e_{0})^{*}$ and (4), to prove (3), it suffices to show that for any $e_{1}\in\mathcal{E}$ with $|e_{1}\cap e_{0}|\geq 2$, there exists $e_{2}\in\mathcal{E}$ with $|e_{2}\cap e_{0}|=1$ such that $e_{2}-e_{0}\subseteq e_{1}-e_{0}$.

Let $e_{1}\in\mathcal{E}$ with $\{u,v\}\subseteq e_{1}\cap e_{0}$, as shown in Figure 3. Observe that $(u,e_{0},v,e_{1},u)$ forms a cycle in $\mathcal{H}$. Since $\mathcal{H}\in\mathcal{L}_{0}$, there exists an edge $e_{2}=\{x,y\}\in\mathcal{E}$ such that $e_{2}\subseteq e_{0}\cup e_{1}$. Moreover, $e_{2}\not\subseteq e_{i}$ for $i\in\{0,1\}$ as $\mathcal{H}$ is Sperner. Without loss of generality, we assume that $x\in e_{1}-e_{0}$ and $y\in e_{0}-e_{1}$, as shown in Figure 3. Clearly, $e_{2}-e_{0}=\{x\}\subseteq e_{1}-e_{0}$, as required. Thus the result holds.

If $\mathcal{H}\in\mathcal{L}_{0}$ is a graph, then it is obvious that $\mathcal{H}-e_{0}\in\mathcal{L}_{0}$ and $(\mathcal{H}/e_{0})^{*}\in\mathcal{L}_{0}$. In the following, we consider the case that $\mathcal{H}$ is not a graph, i.e., there exists at least one edge $e\in\mathcal{E}$ with $|e|>2$.

Lemma 3.2

Let $\mathcal{H}=(V,\mathcal{E})\in\mathcal{L}_{0}$ and $e_{0}\in\mathcal{E}$ with $|e_{0}|>2$. Assume that $\mathcal{H}$ is Sperner. Then, $\mathcal{H}-e_{0}\in\mathcal{L}_{0}$ and $(\mathcal{H}/e_{0})^{*}\in\mathcal{L}_{0}$.

Proof.  It is obvious that $\mathcal{H}-e_{0}\in\mathcal{L}_{0}$ as it is obtained from $\mathcal{H}$ by removing $e_{0}$ directly.

Let $(\mathcal{H}/e_{0})^{*}$ be the hypergraph $(V/e_{0},(\mathcal{E}/e_{0})^{*})$. For any vertex $u$ in $(\mathcal{H}/e_{0})^{*}$, by the definition of contracting, either $u\in V-e_{0}$ or $u=w$, where $w$ is the new vertex in $\mathcal{H}/e_{0}$ after contracting $e_{0}$ in $\mathcal{H}$. For any edge $e^{*}\in(\mathcal{E}/e_{0})^{*}$, by Lemma 3.1, either $e^{*}\in\{e\in\mathcal{E}:e\cap e_{0}=\emptyset\}$ or $e^{*}\in\{(e-e_{0})\cup\{w\}:e\in\mathcal{E},|e\cap e_{0}|=1\}$, implying that $e^{*}$ always has the even size as $\mathcal{H}\in\mathcal{L}_{0}$. Thus, to show that $(\mathcal{H}/e_{0})^{*}\in\mathcal{L}_{0}$, it remains proving the following claim.

Claim: For any cycle $C^{*}=(u_{1},e_{1}^{*},u_{2},e_{2}^{*},\ldots,u_{t},e_{t}^{*},u_{1})$ in $(\mathcal{H}/e_{0})^{*}$, there exist $u,v\in\bigcup_{1\leq i\leq t}e^{*}_{i}$ such that $\{u,v\}\in(\mathcal{E}/e_{0})^{*}$.

To prove this claim, it suffices to show that the claim holds for each minimal cycle $C^{*}$ in $(\mathcal{H}/e_{0})^{*}$, i.e., a cycle that does not contain any other cycle in $(\mathcal{H}/e_{0})^{*}$. In the following, we assume that $C^{*}$ is a minimal cycle in $(\mathcal{H}/e_{0})^{*}$. It is trivial if $|e_{i}^{*}|=2$ for some $i\in\{1,\ldots,t\}$. Thus we also assume that $|e_{i}^{*}|>2$ for all $i=1,\ldots,t$.

By Lemma 3.1, there exist edges $e_{1},\ldots,e_{t}$ in $\mathcal{H}$ with the following properties: for all $i=1,\ldots,t$,

1. (a)

$|e_{i}\cap e_{0}|\leq 1$; 2. (b)

$e_{i}^{*}=e_{i}$ when $e_{i}\cap e_{0}=\emptyset$; and $e_{i}^{*}=(e_{i}-e_{0})\cup\{w\}$ otherwise.

Let $P_{0}=e_{0}\cap\bigcup_{1\leq i\leq t}e_{i}$. Clearly, $\bigcup_{1\leq i\leq t}e_{i}^{*}=\bigcup_{1\leq i\leq t}e_{i}$ when $P_{0}=\emptyset$, and $\bigcup_{1\leq i\leq t}e_{i}^{*}=(\bigcup_{1\leq i\leq t}e_{i}-P_{0})\cup\{w\}$ otherwise.

We are now going to prove the claim on a case-by-case basis.

Case 1: $w\notin\{u_{1},\ldots,u_{t}\}$.

In this case, either $w\notin\bigcup_{1\leq i\leq t}e_{i}^{*}$ or $w\in\bigcup_{1\leq i\leq t}e_{i}^{*}$ but $w\notin\{u_{1},\ldots,u_{t}\}$. Thus there is a cycle $C=(u_{1},e_{1},u_{2},e_{2},\ldots,u_{t},e_{t},u_{1})$ in $\mathcal{H}$. Since $\mathcal{H}\in\mathcal{L}_{0}$, there is an edge $e^{\prime}=\{u,v\}\in\mathcal{E}$ such that $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}$. By Proposition 2.2, $|e^{\prime}\cap e_{0}|\leq 1$.

If $e^{\prime}\cap e_{0}=\emptyset$, by Lemma 3.1, then $e^{\prime}\in(\mathcal{E}/e_{0})^{*}$. Since $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}$ and $e^{\prime}\cap e_{0}=\emptyset$, we have that $e^{\prime}\cap P_{0}=\emptyset$, implying that $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}^{*}$.

If $|e^{\prime}\cap e_{0}|=1$, say $e^{\prime}\cap e_{0}=\{u\}$, then $\{v,w\}\in(\mathcal{E}/e_{0})^{*}$ by Lemma 3.1. Since $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}$ and $e^{\prime}\cap e_{0}=\{u\}$, we have that $P_{0}\neq\emptyset$ which implies $\bigcup_{1\leq i\leq t}e_{i}^{*}=(\bigcup_{1\leq i\leq t}e_{i}-P_{0})\cup\{w\}$. Thus $\{v,w\}\subseteq\bigcup_{1\leq i\leq t}e_{i}^{*}$.

Hence the claim holds in this case.

Case 2: $w\in\{u_{1},\ldots,u_{t}\}$.

Without loss of generality, suppose that $w=u_{1}$. Since $w=u_{1}\in e_{1}^{*}\cap e_{t}^{*}$, by Lemma 3.1, we have $|e_{1}\cap e_{0}|=|e_{t}\cap e_{0}|=1$. By assumtion, $C^{*}$ is a minimal cycle in $({\mathcal{H}}/e_{0})^{*}$, implying that $w\notin e_{i}^{*}$ for all $i=2,\ldots,t-1$, and so $e_{i}\cap e_{0}=\emptyset$ for all $i=2,3,\ldots,t-1$ by Lemma 3.1.

Case 2.1: $e_{1}\cap e_{0}=e_{t}\cap e_{0}=\{x\}$ for some $x\in V$.

In this subcase, $e_{1}=(e_{1}^{*}\setminus\{w\})\cup\{x\}$, $e_{t}=(e_{t}^{*}\setminus\{w\})\cup\{x\}$ and $e_{i}=e_{i}^{*}$ for all $i=2,3,\cdots,t-1$. Thus $C=(x,e_{1},u_{2},e_{2},u_{3},\ldots,u_{t},e_{t},x)$ is a cycle in $\mathcal{H}$, as shown in Figure 4 (i). Since $\mathcal{H}\in\mathcal{L}_{0}$, there is an edge $e^{\prime}=\{u,v\}\in\mathcal{E}$ such that $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}$.

If $x\notin\{u,v\}$, then $e^{\prime}\in(\mathcal{E}/e_{0})^{*}$ and $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}^{*}$.

If $x\in\{u,v\}$, say $x=u$, then $\{w,v\}\in(\mathcal{E}/e_{0})^{*}$ and $\{w,v\}\subseteq\bigcup_{1\leq i\leq t}e_{i}^{*}$.

Thus the claim holds for this subcase.

Case 2.2: $e_{1}\cap e_{0}=\{x\}$ and $e_{t}\cap e_{0}=\{y\}$, where $x,y\in V$ and $x\neq y$.

In this subcase, $e_{1}=(e_{1}^{*}\setminus\{w\})\cup\{x\}$, $e_{t}=(e_{t}^{*}\setminus\{w\})\cup\{y\}$ and $e_{i}=e_{i}^{*}$ for all $i=2,3,\cdots,t-1$. Thus $C=\{y,e_{0},x,e_{1},u_{2},e_{2},u_{3},e_{3},\ldots,u_{t},e_{t},y\}$ is a cycle in $\mathcal{H}$, as shown in Figure 4 (ii). Since $\mathcal{H}\in\mathcal{L}_{0}$, there is an edge $e^{\prime}=\{u,v\}\in\mathcal{E}$ such that

$e^{\prime}\subseteq\bigcup_{0\leq i\leq t}e_{i}$. By Proposition 2.2, $|e^{\prime}\cap e_{0}|\leq 1$.

If $e^{\prime}\cap e_{0}=\emptyset$, then $e^{\prime}\in(\mathcal{E}/e_{0})^{*}$ by Lemma 3.1 and e^{\prime}\subseteq\big{(}\bigcup_{1\leq i\leq t}e_{i}\big{)}\setminus\{x,y\} implying that $e^{\prime}\subseteq\bigcup_{1\leq i\leq t}e_{i}^{*}$.

If $|e^{\prime}\cap e_{0}|=1$, say $e^{\prime}\cap e_{0}=\{u\}$, then $\{w,v\}\in(\mathcal{E}/e_{0})^{*}$ by Lemma 3.1 and v\in\big{(}\bigcup_{1\leq i\leq t}e_{i}\big{)}\setminus\{x,y\}, implying that $\{w,v\}\in\bigcup_{1\leq i\leq t}e_{i}^{*}$.

Thus the claim holds for this subcase.

Therefore, the claim holds in Case 2 and the proof is complete.

4 Proving the main results

Now we are ready to prove Theorems 1.2 and 1.3.

Proof. [Proof of Theorem 1.2] Let $n$ be the number of vertices in $\mathcal{H}$ and $m$ be the number of edges in $\mathcal{H}$. We shall prove the statement by induction on $m$.

When $m=0$, $\mathcal{H}$ is an empty graph and $P(\mathcal{H},\lambda)=\lambda^{n}$. It is obvious that $(-1)^{n}P(\mathcal{H},\lambda)>0$ for any $\lambda\in(-\infty,0)$.

Assume that Theorem 1.2 holds for all hypergraphs in $\mathcal{L}_{0}$ with less than $m$ $(m\geq 1)$ edges. Now let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph in $\mathcal{L}_{0}$, where $|V|=n$ and $|\mathcal{E}|=m$. By (1), we can assume that $\mathcal{H}$ is Sperner.

Let $e_{0}$ be the edge in $\mathcal{H}$ with the largest size. If $|e_{0}|=2$, then $\mathcal{H}$ is a graph and Theorem 1.2 has been proved that it holds for a graph (see [10, 13, 20, 21]).

If $|e_{0}|>2$, by Theorem 2.3 and (1), we have that

[TABLE]

As $\mathcal{H}$ is a Sperner hypergraph in $\mathcal{L}_{0}$, by Lemma 3.2, both $\mathcal{H}-e_{0}$ and $(\mathcal{H}/e_{0})^{*}$ belong to $\mathcal{L}_{0}$. Thus, by the inductive assumption, for any real $\lambda\in(-\infty,0)$,

[TABLE]

As $\mathcal{H}\in\mathcal{L}_{0}$, $|e_{0}|$ is even. Thus, from (5) and (6), for any real $\lambda\in(-\infty,0)$,

[TABLE]

The proof is complete.

For any hypergraph $\mathcal{H}$, let

[TABLE]

Clearly, $Q(\mathcal{H},\lambda)$ is also a polynomial in $\lambda$, as $P(\mathcal{H},\lambda)$ is a polynomial in $\lambda$ and $P(\mathcal{H},0)=0$ (see [26]). To prove Theorem 1.3, it suffices to establish the result below.

Theorem 4.1

If $\mathcal{H}=(V,\mathcal{E})$ is a hypergraph in $\mathcal{L}_{0}^{\prime}$, then for any real $\lambda\in[0,1)$,

[TABLE]

Proof.  Let $\mathcal{E}^{\prime}(\mathcal{H})=\{e\in\mathcal{E}:|e|>2\}$ and $m^{\prime}=|\mathcal{E}^{\prime}(\mathcal{H})|$. We shall prove the statement by induction on $m^{\prime}$. When $m^{\prime}=0$, $\mathcal{H}\in\mathcal{L}^{\prime}_{0}$ implies that $\mathcal{H}$ is a connected graph. Thus (8) holds in this case (see [10, 13]).

Assume that (8) holds for all hypergraphs $\mathcal{H}$ in $\mathcal{L}^{\prime}_{0}$ with $|\mathcal{E}^{\prime}(\mathcal{H})|<m^{\prime}$ ($m^{\prime}\geq 1$).

Now let $\mathcal{H}=(V,\mathcal{E})$ be a Sperner hypergraph in $\mathcal{L}^{\prime}_{0}$ with $|\mathcal{E}^{\prime}(\mathcal{H})|=m^{\prime}$. Let $e_{0}\in\mathcal{E}^{\prime}(\mathcal{H})$. By Theorem 2.3 and (1),

[TABLE]

By (7) and (9),

[TABLE]

As $|e_{0}|>2$ and $\mathcal{H}\in\mathcal{L}^{\prime}_{0}$, $\mathcal{H}-e_{0}\in\mathcal{L}^{\prime}_{0}$ holds.

By Lemma 3.2, $(\mathcal{H}/e_{0})^{*}\in\mathcal{L}_{0}$. Let $\mathcal{H}^{2}$ be the hypergraph $(V,\mathcal{E}_{2})$. As $\mathcal{H}\in\mathcal{L}^{\prime}_{0}$, $\mathcal{H}^{2}$ is connected. By Proposition 2.2, $|e^{\prime}\cap e_{0}|\leq 1$ for each $e^{\prime}\in\mathcal{E}_{2}$. Thus $\mathcal{H}^{2}/e_{0}$ is connected, implying that $(\mathcal{H}/e_{0})^{*}\in\mathcal{L}^{\prime}_{0}$.

Since $|e_{0}|>2$, $|\mathcal{E}^{\prime}(\mathcal{H}-e_{0})|<m^{\prime}$ and $|\mathcal{E}^{\prime}((\mathcal{H}/e_{0})^{*})|<m^{\prime}$. Thus, by inductive assumption, for any real $\lambda\in[0,1)$,

[TABLE]

As $\mathcal{H}\in\mathcal{L}_{0}$, $|e_{0}|$ is even. Thus by (10) and (11), for any real $\lambda\in[0,1)$,

[TABLE]

The proof is complete.

5 Further study

We are not sure if Theorems 1.2 and 1.3 can be extended to larger families of hypergraphs. Especially for Theorem 1.3, we are not sure if its conclusion holds for some connected hypergraph $\mathcal{H}=(V,\mathcal{E})$ whose spanning subhypergraph $(V,\mathcal{E}_{2})$ is not connected.

Problem 5.1

Find a sufficient condition weaker than that in Theorem 1.3 for a hypergraph $\mathcal{H}=(V,\mathcal{E})$ such that $P(\mathcal{H},\lambda)\neq 0$ for all $\lambda\in(0,1)$.

We wonder if Theorems 1.2 and 1.3 can be extended to the chromatic polynomials of mixed hypergraphs. A mixed hypergraph is a triple $\mathcal{H}=(V,\mathcal{C},\mathcal{D})$, where $V$ is a finite set, called the vertex set of $\mathcal{H}$, and $\mathcal{C}$ and $\mathcal{D}$ are collections of subsets of $V$, called the $\mathcal{C}$-edge set and $\mathcal{D}$-edge set respectively. A proper $k$-coloring of $\mathcal{H}=(V,\mathcal{C},\mathcal{D})$ is a mapping $\phi:V\rightarrow\{1,2,\ldots,k\}$ such that $|\{\phi(v):v\in e\}|<|e|$ holds for each $e\in\mathcal{C}$ and $|\{\phi(v):v\in e\}|\geq 2$ holds for each $e\in\mathcal{D}$. The coloring of mixed hypergraphs was introduced by Voloshin [33] in 1995. Since then, the study of colorings of mixed hypergraphs has developed into a separate topic and it has been applied in many fields, such as database management, channel assignments, cyber security [12, 15, 18, 19].

The chromatic polynomial of a mixed hypergraph $\mathcal{H}=(V,\mathcal{C},\mathcal{D})$ is the function, denoted by $P_{mix}(\mathcal{H},\lambda)$, which counts the number of proper $\lambda$-colorings of $\mathcal{H}$ whenever $\lambda$ is a positive integer. Evidently, a weak proper $k$-coloring of a hypergraph $\mathcal{H}=(V,\mathcal{E})$ is actually a proper $k$-coloring of the mixed hypergraph $\mathcal{H}^{\prime}=(V,\emptyset,\mathcal{E})$, implying that $P(\mathcal{H},\lambda)=P_{mix}(\mathcal{H}^{\prime},\lambda)$. Thus, the chromatic polynomials of mixed hypergraphs extend the chromatic polynomials of hypergraphs.

Problem 5.2

Extend Theorems 1.2 and 1.3 to mixed hypergraphs.

Acknowledgment: The authors wish to thank the referees for their very helpful comments and suggestions. The research was partially supported by NTU AcRF project (RP 3/16 DFM) of Singapore.