Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

Abstract

A mixed hypergraph $\mathcal{H}$ is a triple $(X,\mathcal{C},\mathcal{D})$, where X is a finite set and each of $\mathcal{C}$ and $\mathcal{D}$ is a family of subsets of X. For any positive integer $\lambda$, a proper $\lambda$-coloring of $\mathcal{H}$ is an assignment of $\lambda$ colors to vertices in $\mathcal{H}$ such that each member in $\mathcal{C}$ contains at least two vertices assigned the same color and each member in $\mathcal{D}$ contains at least two vertices assigned different colors. The chromatic polynomial of $\mathcal{H}$ is the graph-function counting the number of distinct proper $\lambda$-colorings of $\mathcal{H}$ whenever $\lambda$ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals $(-\infty ,0)$ and $(0,1)$, which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

1. Introduction

The chromatic polynomial of a graph was first introduced by Birkhoff [1] in 1912 with the aim of proving the four-color conjecture. Although this approach has not been realized within the past century, the theory of chromatic polynomials has evolved into an important and independent branch of graph theory, encouraging substantial numbers of studies in this field [2,3,4,5,6,7,8,9,10,11]. In the 1960s, the concept of chromatic polynomials was extended to hypergraphs [12,13,14,15,16,17,18,19,20,21]. In 1993, Voloshin [22] introduced the concept of mixed hypergraphs as well as their chromatic polynomials, which are actually generalizations of chromatic polynomials of graphs and hypergraphs.

The study of the zero distributions of chromatic polynomials is an active research topic [6,7,8,9,10,11,13,14,23]. For graphs, it is well known that chromatic polynomials of graphs have no real zeros in the intervals $(-\infty ,0)$, $(0,1)$ and $(1,32/27]$ (see [6,24,25]), while their zeros are dense in the whole complex plane, as explained by Sokal [26]. Whether these properties can be inherited by hypergraphs and mixed hypergraphs has become a natural research question. Back in 1995, Dohmen [12] showed that for any hypergraph in which each edge has an even size and each cycle contains an edge of size 2, it chromatic polynomial is zero-free in the interval $(-\infty ,0)$. In 2017, the first two authors of this article in [13] proved that chromatic polynomials of hypergraphs have dense real zeros in set of all real numbers, while they [14] recently extended Dohmen’s result to a larger family of hypergraphs and proved that the existence of families of hypergraphs whose chromatic polynomials are zero-free in the intervals $(-\infty ,0)$ and $(0,1)$.

In this article, we further extend the result obtained in [14] on chromatic polynomials of hypergraphs to that on chromatic polynomials of mixed hypergraphs.

2. Preliminaries

In this section, we provide some basic definitions and some known results on chromatic polynomials of mixed hypergraphs.

2.1. Mixed Hypergraphs and Their Chromatic Polynomials

A mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ consists of a vertex setX and two subsets $\mathcal{C}$ and $\mathcal{D}$ of the power set of X. The members in $\mathcal{C}$ and $\mathcal{D}$ are called $\mathcal{C}$-edges and $\mathcal{D}$-edges, respectively. A hypergraph is a mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ with $\mathcal{C}=\varnothing$.

For any positive integer $\lambda$, a proper λ-coloring of a mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ is an assignment of $\lambda$ colors to vertices in $\mathcal{H}$ such that each member in $\mathcal{C}$ contains at least two vertices assigned the same color and each member in $\mathcal{D}$ contains at least two vertices assigned different colors. Thus, the definition of proper $\lambda$-colorings of mixed hypergraphs extends that of hypergraphs. A mixed hypergraph is called colorable if it has at least one proper coloring, and uncolorable otherwise. A simple example of an uncolorable mixed hypergraph is $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$, containing one edge $E\in \mathcal{C}\cap \mathcal{D}$ with $\left|E\right|=2$.

Let $\mathcal{H}$ be a mixed hypergraph and $\lambda$ be a positive integer. The chromatic polynomial of $\mathcal{H}$, denoted by $P(\mathcal{H},\lambda )$, is the function counting the number of proper $\lambda$-colorings of $\mathcal{H}$. Note that $P(\mathcal{H},\lambda )=0$ if $\mathcal{H}$ is uncolorable.

2.2. Known Results on Chromatic Polynomials of Mixed Hypergraphs

In this subsection, we present several known results on chromatic polynomials of mixed hypergraphs, which will be applied in the next sections. The first one shows a family of evidently uncolorable mixed hypergraphs.

Proposition 1

([27]). For any mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$, if there exists $C_0\in \mathcal{C}$ such that $\{x,y\}\in \mathcal{D}$ for every pair of vertices $x,y\in C_0$, then $P(\mathcal{H},\lambda )=0$.

Next, we define an operation on mixed hypergraphs, which is based on the operation of identifying vertices in graphs. Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a mixed hypergraph and $X_0\subset X$. The mixed hypergraph $\mathcal{H}·X_0$ is obtained from $\mathcal{H}$ by removing all $D\in \mathcal{D}$ with $D\subseteq X_0$, removing all $C\in \mathcal{C}$ with $|C\cap X_0|\ge 2$, and identifying all vertices in $X_0$ as one vertex. Thus, $\mathcal{H}·X_0$ is the mixed hypergraph $(X/X_0,\mathcal{C}/X_0,\mathcal{D}/X_0)$, where $X/X_0=(X-X_0)\cup \left\{w\right\}$, w is the new vertex when $\mathcal{H}·X_0$ is produced from $\mathcal{H}$, and

$$\left\{\begin{array}{c}\mathcal{C}/X_0=\{C\in \mathcal{C}:C\cap X_0=\varnothing \}\cup \{(C-X_0)\cup \left\{w\right\}:C\in \mathcal{C},|C\cap X_0|=1\};\hfill \\ \mathcal{D}/X_0=\{D\in \mathcal{D}:D\cap X_0=\varnothing \}\cup \{(D-X_0)\cup \left\{w\right\}:D\in \mathcal{D},D\cap X_0\ne \varnothing ,D⊈X_0\}.\hfill \end{array}\right.$$

(1)

For an edge $E\in \mathcal{C}\cup \mathcal{D}$, let $\mathcal{H}/E$ denote the mixed hypergraph $(\mathcal{H}-E)·E$, where $\mathcal{H}-E=(X,\mathcal{C}-\{E\},\mathcal{D}-\{E\left\}\right)$. We also say that $\mathcal{H}/E$ is obtained from $\mathcal{H}$ by contracting the edge E.

Clearly, for any colorable mixed hypergraph $\mathcal{H}$ with a $\mathcal{C}$-edge C of size two, $P(\mathcal{H},\lambda )=P(\mathcal{H}/C,\lambda )$. Therefore, in the following, we need only to consider mixed hypergraphs $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ satisfying $\left|C\right|\ge 3$ for all $C\in \mathcal{C}$.

The study of chromatic polynomials of mixed hypergraphs can be restricted to a special class of mixed hypergraphs. A mixed hypergraph is called reduced if no edge is a subset of any other edge of the same type and the size of each $\mathcal{C}$-edge is at least 3. For any mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$, we define the reduced subhypergraph ${\mathcal{H}}^{*}$ to be the maximal reduced subhypergraph among all the subhypergraphs of $\mathcal{H}$. In other words, ${\mathcal{H}}^{*}$ is obtained from $\mathcal{H}$ by removing any edge $C\in \mathcal{C}$ or $D\in \mathcal{D}$ repeatedly whenever $C^{\prime }\subseteq C$ or $D^{\prime }\subseteq D$ holds for another edge $C^{\prime }\in \mathcal{C}$ or $D^{\prime }\in \mathcal{D}$.

It is not difficult to show that ${\mathcal{H}}^{*}$ is uniquely determined by $\mathcal{H}$, and if a mixed hypergraph $\mathcal{H}$ is reduced, then ${\mathcal{H}}^{*}=\mathcal{H}$. Moreover, the following proposition ensures that we only need to focus on reduced mixed hypergraphs when studying chromatic polynomials of mixed hypergraphs.

Proposition 2

([27]). Let $\mathcal{H}$ be a mixed hypergraph and ${\mathcal{H}}^{*}$ be the reduced subhypergraph of $\mathcal{H}$. Then $P(\mathcal{H},\lambda )=P({\mathcal{H}}^{*},\lambda )$.

Next, the deletion-contraction formula for chromatic polynomials of graphs and hypergraphs can be extended directly to chromatic polynomials of mixed hypergraphs as follows.

Theorem 1.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a colorable and reduced mixed hypergraph with $\mathcal{D}\ne \varnothing$. For any $D\in \mathcal{D}$,

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

(2)

where $\mathcal{H}-D=(X,\mathcal{C},\mathcal{D}-\{D\left\}\right)$.

An alternative form of Theorem 1 was given by Voloshin [27].

Theorem 2

([27]). Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a colorable and reduced mixed hypergraph. For any two vertices $x,y\in X$ with $\{x,y\}\notin \mathcal{D}$, the chromatic polynomial of $\mathcal{H}$ is

$$P(\mathcal{H},\lambda )=P(\mathcal{H}{+}_d\{x,y\},\lambda )+P(\mathcal{H}·\{x,y\},\lambda ),$$

(3)

where $\mathcal{H}{+}_d\{x,y\}=(X,\mathcal{C},\mathcal{D}\cup \left\{\{x,y\}\right\})$ is a mixed hypergraph obtained from $\mathcal{H}$ by adding a new $\mathcal{D}$-edge $\{x,y\}$.

By applying Theorem 2 repeatedly, we get the following result, which will be used to prove our main results in Section 5.

Corollary 1.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a colorable and reduced mixed hypergraph with $\mathcal{C}\ne \varnothing$ and let $C_0\in \mathcal{C}$ with $|C_0|\ge 3$. Suppose that $\{e_1,e_2,\dots ,e_r\}=\{\{u,v\}\subseteq C_0:u\ne v,\{u,v\}\notin \mathcal{D}\}$. Then $r\ge 1$ and

$$P(\mathcal{H},\lambda )=\sum _{i=0}^{r-1}P({\mathcal{H}}_i·e_{i+1},\lambda ),$$

(4)

where ${\mathcal{H}}_0=\mathcal{H}$ and ${\mathcal{H}}_i={\mathcal{H}}_{i-1}{+}_d{e}_i$ for $i=1,2,\dots ,r-1$.

Proof. 

As ${\mathcal{H}}_0=\mathcal{H}$ and ${\mathcal{H}}_i={\mathcal{H}}_{i-1}{+}_d{e}_i$ for $i=1,2,\dots ,r-1$, we can write ${\mathcal{H}}_i$ as $(X,\mathcal{C},{\mathcal{D}}_i)$ for $i=1,2,\dots ,r-1$, where ${\mathcal{D}}_i=\mathcal{D}\cup \{e_1,e_2,\dots ,e_i\}$.

For each $i=0,1,\dots ,r-1$, we apply Theorem 2 on ${\mathcal{H}}_i$ and obtain that

$$P(\mathcal{H},\lambda )=P({\mathcal{H}}_r,\lambda )+\sum _{i=0}^{r-1}P({\mathcal{H}}_i·e_{i+1},\lambda ),$$

(5)

where ${\mathcal{H}}_r={\mathcal{H}}_{r-1}{+}_d{e}_r$. Clearly, ${\mathcal{H}}_r$ contains a $\mathcal{C}$-edge $C_0$ in which every two-element subset $\{u,v\}$ of $C_0$ is a $\mathcal{D}$-edge. Then by Proposition 1, ${\mathcal{H}}_r$ is uncolorable and so $P({\mathcal{H}}_r,\lambda )=0$ for any positive integer $\lambda$. The proof is complete. □

3. Main Results

In this section, we present our main results after defining some concepts regarding mixed hypergraphs that are necessary to derive the results of this paper.

A cycle of a mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ is defined to be a sequence of alternating vertices and edges $\mathbb{C}=(v_1,E_1,v_2,E_2,\dots ,v_t,E_t,v_1),$ where $t\ge 2$, $v_1,v_2,\dots ,v_t$ are pairwise distinct vertices and $E_i\ne E_j$ whenever $E_i\notin \mathcal{C}\cap \mathcal{D}$ or $E_j\notin \mathcal{C}\cap \mathcal{D}$ for all $1\le i<j\le t$, such that $\{v_i,v_{i+1}\}\subseteq E_i$ for all $i=1,2,\dots ,t$, where $v_{t+1}=v_1$. Note that it is possible that $E_i=E_j$ for some $1\le i<j\le t$. In such case, by the definition of a cycle, $E_i=E_j\in \mathcal{C}\cap \mathcal{D}$. A cycle $\mathbb{C}=(v_1,E_1,v_2,E_2,\dots ,v_t,E_t,v_1)$ is called a $\mathcal{D}$-cycle if $E_i\in \mathcal{D}\setminus \mathcal{C}$ for all $i=1,2,\dots ,t$.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a mixed hypergraph and $\mathcal{E}\subseteq \{E\subseteq X:|E|\ge 1\}$. A subset $X_0$ of X is said to be $\mathcal{E}$-independent if $E⊈X_0$ holds for every $E\in \mathcal{E}$. Let ${\mathcal{D}}_2\left(\mathcal{H}\right)=\{D\in \mathcal{D}:|D|=2\}$. Sometimes, ${\mathcal{D}}_2\left(\mathcal{H}\right)$ is written as ${\mathcal{D}}_2$ for short. Then, $X_0\subseteq X$ is not ${\mathcal{D}}_2$-independent in $\mathcal{H}$ if and only if $D\subseteq X_0$ holds for some $D\in {\mathcal{D}}_2$.

Listed below are four conditions (I)–(IV) concerning a mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$. Our main results of this paper are subject to some or all of them:

(I)

$\left|D\right|$ is even for each $D\in \mathcal{D}$;

(II)

For every $\mathcal{D}$-cycle $(v_1,D_1,v_2,D_2,\dots ,v_t,D_t,v_1)$ of $\mathcal{H}$, the set ${\bigcup }_{1\le i\le t}{D}_i$ is not ${\mathcal{D}}_2$-inde pendent in $\mathcal{H}$;

(III)

For every cycle $(v_1,E_1,v_2,E_2,\dots ,v_t,E_t,v_1)$ of $\mathcal{H}$, which is not a $\mathcal{D}$-cycle, the set $\{v_1,\dots ,v_t\}$ is not ${\mathcal{D}}_2$-independent in $\mathcal{H}$; and

(IV)

The spanning subhypergraph ${\mathcal{H}}_2=(X,{\mathcal{D}}_2)$ of $\mathcal{H}$ is connected.

Let ${\mathcal{L}}_0$ be the set of hypergraphs satisfying conditions (I) and (II) and ${\mathcal{L}}_0^{\prime }$ be the set of hypergraphs satisfying conditions (I), (II), and (IV). In [14], the first two authors of this article obtained the following results on hypergraphs in the two sets ${\mathcal{L}}_0$ and ${\mathcal{L}}_0^{\prime }$.

Theorem 3

([14]). For any hypergraph $\mathcal{H}$ in ${\mathcal{L}}_0$, $P(\mathcal{H},\lambda )$ is zero-free in the interval $(-\infty ,0)$; Furthermore, if $\mathcal{H}$ is in ${\mathcal{L}}_0^{\prime }$, then $P(\mathcal{H},\lambda )$ is zero-free in the interval $(0,1)$ and $P(\mathcal{H},\lambda )$ has no multiple zero at $\lambda =0$.

In this article, we extend the above results to chromatic polynomials of mixed hypergraphs. Let ${\mathcal{Q}}_0$ be the set of mixed hypergraphs $\mathcal{H}$ with ${\mathcal{H}}^{*}$ satisfying conditions (I)–(III) and ${\mathcal{Q}}_0^{\prime }$ be the set of mixed hypergraphs $\mathcal{H}$ with ${\mathcal{H}}^{*}$ satisfying conditions (I)–(IV). Clearly, ${\mathcal{L}}_0\subset {\mathcal{Q}}_0$ and ${\mathcal{L}}_0^{\prime }\subset {\mathcal{Q}}_0^{\prime }$. We prove the following two results on the chromatic polynomials of mixed hypergraphs in ${\mathcal{Q}}_0$ and ${\mathcal{Q}}_0^{\prime }$.

Theorem 4.

For any colorable mixed hypergraph $\mathcal{H}$ in ${\mathcal{Q}}_0$, $P(\mathcal{H},\lambda )$ is zero-free in the interval $(-\infty ,0)$.

Observe that in Figure 1 the hypergraph ${\mathcal{H}}_1$ does not belong to ${\mathcal{Q}}_0$, while the hypergraph ${\mathcal{H}}_2$ belongs to ${\mathcal{Q}}_0$. By applying Corollary 1 and Theorem 1 repeatedly, it can be calculated that

$$P({\mathcal{H}}_1,\lambda )={\lambda }^3+4{\lambda }^2-4\lambda ,$$

which has a real zero at $-2-2\sqrt{2}\approx -4.828$, and

$$P({\mathcal{H}}_2,\lambda )=4\lambda (\lambda -1),$$

which shows that $P({\mathcal{H}}_2,\lambda )$ has no real zeros in the interval $(-\infty ,0)$.

Theorem 5.

For any colorable mixed hypergraph $\mathcal{H}$ in ${\mathcal{Q}}_0^{\prime }$, $P(\mathcal{H},\lambda )$ is zero-free in the interval $(0,1)$. Furthermore, $P(\mathcal{H},\lambda )$ has no multiple zero at $\lambda =0$.

Theorems 4 and 5 imply Theorem 3 directly, as ${\mathcal{L}}_0\subset {\mathcal{Q}}_0$ and ${\mathcal{L}}_0^{\prime }\subset {\mathcal{Q}}_0^{\prime }$. The proofs of Theorems 4 and 5 are given in Section 5.

4. ${\mathcal{Q}}_{\mathbf{0}}$ Is Deletion and Contraction Closable

In this section, we show that ${\mathcal{Q}}_0$ is deletion and contraction closable; i.e., for any mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ with $\mathcal{C}\cup \mathcal{D}\ne \varnothing$, at least one of the following cases happens:

(1)

There exists some $D_0\in \mathcal{D}$ such that both $\mathcal{H}-D_0\in {\mathcal{Q}}_0$ and ${(\mathcal{H}/D_0)}^{*}\in {\mathcal{Q}}_0$ hold;

(2)

There exists some $C_0\in \mathcal{C}$ such that both $\mathcal{H}-C_0\in {\mathcal{Q}}_0$ and ${(\mathcal{H}·X_0)}^{*}\in {\mathcal{Q}}_0$ hold, where $X_0\subset C_0$ with $|X_0|=2$ and $X_0\notin \mathcal{D}$.

Let us first study the structure of a mixed hypergraph which satisfies condition (III).

Lemma 1.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a reduced mixed hypergraph satisfying condition (III). Then,

(i)

For any distinct pair $C_1,C_2\in \mathcal{C}$ with $|C_1\cap C_2|\ge 2$, $\{\{u,v\}:u,v\in C_1\cap C_2\}\subseteq \mathcal{D}$ holds;

(ii)

For any $C\in \mathcal{C}$ and $D\in \mathcal{D}$ with $\left|D\right|>2$, $|C\cap D|\le 1$ holds; and

(iii)

If $\mathcal{H}$ is colorable, then $\mathcal{C}\cap \mathcal{D}=\varnothing$.

Proof. 

(i)

Suppose that there exist two vertices $u_0,v_0\in C_1\cap C_2$ with $\{u_0,v_0\}\notin \mathcal{D}$. By definition, $(u_0,C_1,v_0,C_2,u_0)$ forms a cycle of $\mathcal{H}$. Evidently, $(u_0,C_1,v_0,C_2,u_0)$ is not a $\mathcal{D}$-cycle of $\mathcal{H}$. As $\mathcal{H}$ satisfies condition (III), $\{u_0,v_0\}\in \mathcal{D}$, a contradiction.

(ii)

Assume that there exist $C_0\in \mathcal{C}$ and $D_0\in \mathcal{D}$ with $|D_0|>2$ such that $|C_0\cap D_0|\ge 2.$ Let $u,v\in C_0\cap D_0$, where $u,v\in X$ and $u\ne v$. By definition, $(u,C_0,v,D_0,u)$ forms a cycle, but not a $\mathcal{D}$-cycle, of $\mathcal{H}$. Furthermore, as $\mathcal{H}$ satisfies condition (III), $\{u,v\}\in \mathcal{D}$. As $D_0\in \mathcal{D}$ with $|D_0|>2$, $\{u,v\}\subset D_0$, implying that $\mathcal{H}$ is not reduced, a contradiction.

(iii)

Assume that there exists $E_0\in \mathcal{C}\cap \mathcal{D}$. By the definition of a proper coloring of a mixed hypergraph, if $\mathcal{H}$ is colorable and $E_0\in \mathcal{C}\cap \mathcal{D}$, then $|E_0|\ge 3$. From the result (ii) above, we have $|E_0|\le 1$, a contradiction. □

For any colorable and reduced mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$, by Lemma 1 (iii), $\mathcal{C}\cap \mathcal{D}=\varnothing$ holds.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a reduced mixed hypergraph and $X_0\subseteq X$ with $|X_0|\ge 2$. It is possible that $\mathcal{H}·X_0$ is not reduced. By Proposition 2, we consider ${(\mathcal{H}·X_0)}^{*}$ instead of $\mathcal{H}·X_0$ in this paper. As $\mathcal{H}/X_0=(\mathcal{H}-X_0)·X_0$ when $X_0\in \mathcal{D}$, in what follows, we write both $\mathcal{H}·X_0$ and $\mathcal{H}/X_0$ as $(X/X_0,\mathcal{C}/X_0,\mathcal{D}/X_0)$ and by the notation for the reduced mixed subhypergraphs, both ${(\mathcal{H}·X_0)}^{*}$ and ${(\mathcal{H}/X_0)}^{*}$ are written as $(X/X_0,{(\mathcal{C}/X_0)}^{*},{(\mathcal{D}/X_0)}^{*})$. Hence, ${(\mathcal{D}/X_0)}^{*}\subseteq \mathcal{D}/X_0$.

Lemma 2.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ be a reduced mixed hypergraph and $X_0\subseteq X$ with $|X_0|\ge 2$. If $D\in \mathcal{D}/X_0$ with $\left|D\right|=2$, then $D\in {(\mathcal{D}/X_0)}^{*}$.

Proof. 

By the definition of $\mathcal{H}·X_0$,

$$\mathcal{D}/X_0=\{D\in \mathcal{D}:D\cap X_0=\varnothing \}\cup \{(D-X_0)\cup \left\{w\right\}:D\in \mathcal{D},D\cap X_0\ne \varnothing ,D⊈X_0\},$$

(6)

where $w\notin X$. Suppose that there exists $D_0\in \mathcal{D}/X_0$ with $|D_0|=2$ but $D_0\notin {(\mathcal{D}/X_0)}^{*}$. By the definition of the reduced mixed subhypergraph, there must be an edge $D_1\in {(\mathcal{D}/X_0)}^{*}$ such that $D_1\subset D_0$, implying $|D_1|=1$ as $|D_0|=2$. As $\mathcal{H}$ is reduced, $D_1\notin \mathcal{D}$. By (6), there is an edge $D_1^{\prime }\in \mathcal{D}$ such that $D_1=(D_1^{\prime }-X_0)\cup \left\{w\right\}$. As $|D_1|=1$, $D_1^{\prime }\subseteq X_0$ in $\mathcal{H}$. However, if $D_1^{\prime }\subseteq X_0$, by (6) again, then $D_1\notin \mathcal{D}/X_0$, implying that $D_1\notin {(\mathcal{D}/X_0)}^{*}$, a contradiction. □

Lemma 3.

If $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ is a reduced mixed hypergraph satisfying condition (II), then for any $D_0\in \mathcal{D}$ with $|D_0|>2$,

$${(\mathcal{D}/D_0)}^{*}\subseteq \{D\in \mathcal{D}:D\cap D_0=\varnothing \}\cup \{(D-D_0)\cup \left\{w\right\}:|D\cap D_0|=1,D\in \mathcal{D}-D_0\}$$

(7)

holds, where $w\notin X$.

Proof. 

By the definition of $\mathcal{H}/D_0$, we have that

$$\mathcal{D}/D_0=\{D\in \mathcal{D}:D\cap D_0=\varnothing \}\cup \{(D-D_0)\cup \left\{w\right\}:D\in \mathcal{D},D\cap D_0\ne \varnothing \}.$$

(8)

Thus, it suffices to show that for any $D_1\in \mathcal{D}$ with $|D_1\cap D_0|\ge 2$, there exists $D_2\in {\mathcal{D}}_2$ with $|D_2\cap D_0|=1$ such that $D_2-D_0\subseteq D_1-D_0$.

Let $D_1\in \mathcal{D}$ with $\{u,v\}\subseteq D_1\cap D_0$. Observe that $(u,D_1,v,D_0,u)$ forms a $\mathcal{D}$-cycle in $\mathcal{H}$. As $\mathcal{H}$ satisfies condition (II), there exist two distinct vertices $x,y\in D_0\cup D_1$ such that $D_2=\{x,y\}\in \mathcal{D}$. As $\mathcal{H}$ is a reduced mixed hypergraph, $D_2⊈D_i$ for both $i=0,1$, implying, without loss of generality, that $x\in D_0-D_1$ and $y\in D_1-D_0$. Thus, $D_2-D_0\subseteq D_1-D_0$. □

Now we are ready to show that ${\mathcal{Q}}_0$ is deletion and contraction closable. In [14], the authors of this article proved that for any reduced mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ with $\mathcal{C}=\varnothing$, $\mathcal{H}-D\in {\mathcal{Q}}_0$ and ${(\mathcal{H}/D)}^{*}\in {\mathcal{Q}}_0$; i.e., ${\mathcal{L}}_0\subset {\mathcal{Q}}_0$ is deletion and contraction closable. When $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ with $\mathcal{C}\ne \varnothing$, we shall show that ${\mathcal{Q}}_0$ is also deletion and contraction closable by completing the following lemma.

Lemma 4.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ with $\mathcal{C}\ne \varnothing$ and $X_0\subset C_0\in \mathcal{C}$ with $|X_0|=2$. Assume that $\mathcal{H}$ is a reduced mixed hypergraph and $X_0\notin \mathcal{D}$. Then, $\mathcal{H}-C_0\in {\mathcal{Q}}_0$ and ${(\mathcal{H}·X_0)}^{*}\in {\mathcal{Q}}_0$.

Proof. 

Since $\mathcal{H}\in {\mathcal{Q}}_0$ and $\mathcal{H}-C_0=(X,\mathcal{C}-\left\{C_0\right\},\mathcal{D})$, it is clear that $\mathcal{H}-C_0\in {\mathcal{Q}}_0$. To show that ${(\mathcal{H}·X_0)}^{*}\in {\mathcal{Q}}_0$ is equivalent to showing that ${(\mathcal{H}·X_0)}^{*}$ satisfies conditions (I)–(III).

• Claim 1:${(\mathcal{H}·X_0)}^{*}$ satisfies condition (I).

By the definition of $\mathcal{H}·X_0$, we have that

$$\mathcal{D}/X_0=\{D\in \mathcal{D}:D\cap X_0=\varnothing \}\cup \{(D-X_0)\cup \left\{w\right\}:D\in \mathcal{D},D\cap X_0\ne \varnothing ,D⊈X_0\}.$$

(9)

As $\mathcal{H}\in {\mathcal{Q}}_0$, $X_0\subset C_0\in \mathcal{C}$ and $X_0\notin \mathcal{D}$, by Lemma 1 (ii), $|X_0\cap D|\le 1$ holds for all $D\in \mathcal{D}$. Thus, (9) can be rewritten as below:

$$\mathcal{D}/X_0=\{D\in \mathcal{D}:D\cap X_0=\varnothing \}\cup \{(D-X_0)\cup \left\{w\right\}:D\in \mathcal{D},|D\cap X_0|=1\}.$$

(10)

By the definition of ${(\mathcal{H}·X_0)}^{*}$, ${(\mathcal{D}/X_0)}^{*}\subseteq \mathcal{D}/X_0$, implying that ${(\mathcal{H}·X_0)}^{*}$ satisfies condition (I) as $\mathcal{H}\in {\mathcal{Q}}_0$.

• Claim 2:${(\mathcal{H}·X_0)}^{*}$ satisfies conditions (II) and (III).

By definition, $X/X_0=(X-X_0)\cup \left\{w\right\},$ where $w\notin X$, and

$$\left\{\begin{array}{c}{(\mathcal{C}/X_0)}^{*}\subseteq \{C\in \mathcal{C}:C\cap X_0=\varnothing \}\cup \{(C-X_0)\cup \left\{w\right\}:C\in \mathcal{C},|C\cap X_0|=1\};\hfill \\ {(\mathcal{D}/X_0)}^{*}\subseteq \{D\in \mathcal{D}:D\cap X_0=\varnothing \}\cup \{(D-X_0)\cup \left\{w\right\}:D\in \mathcal{D},|D\cap X_0|=1\}.\hfill \end{array}\right.$$

(11)

If ${(\mathcal{H}·X_0)}^{*}$ contains no cycles, by Claim 1, ${(\mathcal{H}·X_0)}^{*}\in {\mathcal{Q}}_0$. Now we assume that ${(\mathcal{H}·X_0)}^{*}$ contains cycles. Let ${\mathbb{C}}^{*}=(v_1,E_1^{*},v_2,E_2^{*},\dots ,v_t,E_t^{*},v_1)$ be any cycle in ${(\mathcal{H}·X_0)}^{*}$, where $v_i\in X/X_0$ and $E_i^{*}\in {(\mathcal{C}/X_0)}^{*}\cup {(\mathcal{D}/X_0)}^{*}$ for each $i=1,2,\cdots ,t$. We shall complete the proof of Claim 2 by the following two sub-claims:

• Claim 2.1: If ${\mathbb{C}}^{*}$ is a $\mathcal{D}$-cycle, then there exists an edge $\{x,y\}\in {(\mathcal{D}/X_0)}^{*}$ such that

  $$\{x,y\}\subseteq \bigcup _{1\le i\le t}{E}_i^{*}.$$

• Claim 2.2: If ${\mathbb{C}}^{*}$ is not a $\mathcal{D}$-cycle, then there exists an edge $\{v_i,v_j\}\in {(\mathcal{D}/X_0)}^{*}$ for some $1\le i<j\le t$.

From (11), there exist edges $E_1,E_2,\dots ,E_t$ in $\mathcal{H}$ with the following properties: for all $i=1,2,\dots ,t$,

(a)

$E_i\in \mathcal{C}$ if and only if $E_i^{*}\in {(\mathcal{C}/X_0)}^{*}$;

(b)

$|E_i\cap X_0|\le 1$;

(c)

$E_i^{*}=E_i$ when $E_i\cap X_0=\varnothing$ and $E_i^{*}=(E_i-X_0)\cup \left\{w\right\}$ otherwise.

Now we are going to prove Claims 2.1 and 2.2 on a case-by-case basis.

• Case 1:$w\notin \{v_1,v_2,\dots ,v_t\}$.

In this case, $\mathbb{C}=(v_1,E_1,v_2,E_2,\dots ,v_t,E_t,v_1)$ is a cycle in $\mathcal{H}$.

If ${\mathbb{C}}^{*}$ is a $\mathcal{D}$-cycle in ${(\mathcal{H}·X_0)}^{*}$, then $E_i^{*}\in {(\mathcal{D}/X_0)}^{*}$ for all $i=1,2,\dots ,t$, implying that $E_i\in \mathcal{D}$ for all $i=1,2,\dots ,t$. Thus, $\mathbb{C}$ is a $\mathcal{D}$-cycle in $\mathcal{H}$ and there is an edge $D^{\prime }=\{x,y\}\in {\mathcal{D}}_2$ such that $D^{\prime }\subseteq {\bigcup }_{1\le i\le t}{E}_i$. As $X_0\subset C_0\in \mathcal{C}$, by Lemma 1 (ii), $|D^{\prime }\cap X_0|\le 1$. If $D^{\prime }\cap X_0=\varnothing$, by (11) and Lemma 2, $D^{\prime }=\{x,y\}\in {(\mathcal{D}/X_0)}^{*}$ and $D^{\prime }\subseteq {\bigcup }_{1\le i\le t}{E}_i^{*}$. If $|D^{\prime }\cap X_0|=1$, then either $x\in X_0$ or $y\in X_0$. Without loss of generality, we assume that $D^{\prime }\cap X_0=\left\{x\right\}$. Thus, $x\in {\bigcup }_{1\le i\le t}{E}_i$ and $y\notin X_0$. By (11) and Lemma 2, there is an edge

$$D^{*}=(D^{\prime }-X_0)\cup \left\{w\right\}=\{y,w\}\in {(\mathcal{D}/X_0)}^{*},$$

where $D^{*}=\{y,w\}\subseteq {\bigcup }_{1\le i\le t}{E}_i^{*}$. Thus, Claim 2.1 holds in this case.

If ${\mathbb{C}}^{*}$ is not a $\mathcal{D}$-cycle in ${(\mathcal{H}·X_0)}^{*}$, then there exists $E_j^{*}\in {(\mathcal{C}/X_0)}^{*}$ for some $j\in \{1,2,\dots ,t\}$, implying that $E_j\in \mathcal{C}$. Thus, $\mathbb{C}$ is not a $\mathcal{D}$-cycle in $\mathcal{H}$ and there is an edge $D^{\prime }=\{v_i,v_j\}\in {\mathcal{D}}_2$ for some $1\le i<j\le t$. As $w\notin \{v_1,v_2,\dots ,v_t\}$ in this subcase, $D^{\prime }\cap X_0=\varnothing$. By (11) and Lemma 2, $D^{\prime }=\{v_i,v_j\}\in {(\mathcal{D}/X_0)}^{*}$. Thus, Claim 2.2 holds in this case.

• Case 2:$w\in \{v_1,v_2,\dots ,v_t\}$.

Without loss of generality, suppose that $w=v_1$. As $w=v_1\in E_1^{*}\cap E_t^{*}$, by (11), we have $|E_1\cap X_0|=|E_t\cap X_0|=1$, while $E_i\cap X_0=\varnothing$ for all $i=2,3,\dots ,t-1$.

• Case 2.1:$E_1\cap X_0=E_t\cap X_0=\left\{v\right\}$ for some $v\in X_0$.

Clearly, $\mathbb{C}=(v,E_1,v_2,E_2,\dots ,v_t,E_t,v)$ is a cycle in $\mathcal{H}$. Thus, it can be proved similarly as in Case 1 that Claims 2.1 and 2.2 hold in this subcase.

• Case 2.2:$E_1\cap X_0=\left\{u\right\}$ and $E_t\cap X_0=\left\{v\right\}$, where $u\ne v$.

By (b), $E_i\ne C_0$ for all $i=1,\cdots ,t$. As $X_0\subset C_0$, $u\in E_1\cap C_0$ and $v\in E_t\cap C_0$,

$$\mathbb{C}=(v,C_0,u,E_1,v_2,E_2,v_3,\dots ,v_t,E_t,v)$$

is a cycle, but not a $\mathcal{D}$-cycle, in $\mathcal{H}$ as $C_0\in \mathcal{C}$. As $\mathcal{H}\in {\mathcal{Q}}_0$, there is an edge $D^{\prime }\in {\mathcal{D}}_2$ such that $D^{\prime }\subseteq \{u,v,v_1,v_2,\dots ,v_t\}$. Furthermore, as $X_0\subset C_0\in \mathcal{C}$, $|D^{\prime }\cap X_0|\le 1$, which implies that $|\{u,v\}\cap D^{\prime }|\le 1$.

If $\{u,v\}\cap D^{\prime }=\varnothing$, by (11) and Lemma 2, then $D^{\prime }\in {(\mathcal{D}/X_0)}^{*}$ and $D^{\prime }\subseteq \{v_2,v_3,\dots ,v_t\}$$\subset {\bigcup }_{1\le i\le t}{E}_i^{*}$. Thus, Claims 2.1 and 2.2 hold.

If $|\{u,v\}\cap D^{\prime }|=1$, then either $u\in D^{\prime }$ or $v\in D^{\prime }$. Without loss of generality, we assume that $u\in D^{\prime }$ and $D^{\prime }=\{u,v_k\}$ for some $k\in \{2,3,\dots ,t\}$. Thus, $D^{\prime }\cap X_0=\left\{u\right\}$ and $v_k\notin X_0$. By (11) and Lemma 2, there is an edge

$$D^{*}=(D^{\prime }-X_0)\cup \left\{w\right\}=\{v_k,w\}\in {(\mathcal{D}/X_0)}^{*}.$$

As $w\in \{v_1,v_2,\dots ,v_t\}$ in this case, $D^{*}\subseteq \{v_1,v_2,\dots ,v_t\}\subset {\bigcup }_{1\le i\le t}{E}_i^{*}$, implying that Claims 2.1 and 2.2 hold. □

5. Proofs of Theorems 4 and 5

In this section, we give the proofs of Theorems 4 and 5 after presenting the following result, which will be applied to prove Theorem 4.

Lemma 5.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ with $\mathcal{C}\ne \varnothing$. Assume that $\mathcal{H}$ is reduced. For any $C_0\in \mathcal{C}$ and $\{u_1,u_2\}\subseteq C_0$ with $\{u_1,u_2\}\notin \mathcal{D}$,

(i)

If there exist distinct $\mathcal{C}$-edges $C_1,C_2\in \mathcal{C}-\left\{C_0\right\}$ with $C_i\cap C_0=\left\{u_i\right\}$ for $i=1,2$, and $\varnothing \ne C_1-\left\{u_1\right\}\subseteq C_2-\left\{u_2\right\}$, then ${\mathcal{H}}^{\mathcal{C}-C_0}·\{u_1,u_2\}$ is uncolorable, where ${\mathcal{H}}^{\mathcal{C}-C_0}=(X,\mathcal{C}-\left\{C_0\right\},\mathcal{D})$;

(ii)

If ${({\mathcal{H}}^{\mathcal{C}-C_0}·\{u_1,u_2\})}^{*}=(X^{*},{\mathcal{C}}^{*},{\mathcal{D}}^{*})$ is colorable, then

$${\mathcal{C}}^{*}=\{C\in \mathcal{C}:C\cap e_0=\varnothing \}\cup \{(C-e_0)\cup \left\{w\right\}:C\in \mathcal{C},|C\cap e_0|=1\},$$

(12)

where $e_0=\{u_1,u_2\}$. Moreover, $|{\mathcal{C}}^{*}|=|\mathcal{C}|-1$.

Proof. 

(i)

Let $e_0=\{u_1,u_2\}$ and $A_0=C_1-e_0$. Thus,

$$A_0\subseteq C_1\cap C_2,C_1=A_0\cup \left\{u_1\right\}\mathrm{and}{A}_0\cup \left\{u_2\right\}\subseteq C_2.$$

As $\mathcal{H}\in {\mathcal{Q}}_0$, by Lemma 1 (i), $\{v_1,v_2\}\in \mathcal{D}$ for each pair $v_1,v_2\in A_0$. Moreover, $(v,C_1,u_1,C_0,u_2,C_2,v)$ is a cycle in $\mathcal{H}$, where v is any vertex in $A_0$. As $\mathcal{H}\in {\mathcal{Q}}_0$ and $e_0=\{u_1,u_2\}\notin \mathcal{D}$, for each $v\in A_0$, $\{v,u_j\}\in \mathcal{D}$ holds for some $j\in \{1,2\}$.

Let ${(\mathcal{H}·e_0)}^{*}=(X^{*},{\mathcal{C}}^{*},{\mathcal{D}}^{*})$. As $e_0\notin \mathcal{D}$, by Lemma 1 (ii), $|D\cap e_0|\le 1$ for each $D\in \mathcal{D}$. By the definition of $\mathcal{H}·e_0$, $X^{*}=(X-e_0)\cup \left\{w\right\}$, where $w\notin X$, and

$$\left\{\begin{array}{c}{\mathcal{C}}^{*}\subseteq \{C\in \mathcal{C}:C\cap e_0=\varnothing \}\cup \{(C-e_0)\cup \left\{w\right\}:C\in \mathcal{C},|C\cap e_0|=1\};\hfill \\ {\mathcal{D}}^{*}\subseteq \{D\in \mathcal{D}:D\cap e_0=\varnothing \}\cup \{(D-e_0)\cup \left\{w\right\}:D\in \mathcal{D},|D\cap e_0|=1\}.\hfill \end{array}\right.$$

(13)

As $\mathcal{H}$ is reduced, $C_1\cap e_0=\left\{u_1\right\}$ and $C_1=A_0\cup \left\{u_1\right\}$,

$$A_0\cup \left\{w\right\}=(C_1-e_0)\cup \left\{w\right\}\in {\mathcal{C}}^{*}.$$

As for any $v\in A_0$, either $\{v,u_1\}\in \mathcal{D}$ or $\{v,u_2\}\in \mathcal{D}$, by (13), $\{v,w\}\in {\mathcal{D}}^{*}$ for each $v\in A_0$. As $\{v_1,v_2\}\in \mathcal{D}$ for each pair $v_1,v_2\in A_0$ and $A_0\cap e_0=\varnothing$, by (13), $\{v_1,v_2\}\in {\mathcal{D}}^{*}$ for each pair $v_1,v_2\in A_0$. Thus, by Proposition 1, ${({\mathcal{H}}^{\mathcal{C}-C_0}·e_0)}^{*}$ is uncolorable.

(ii)

Let $e_0=\{u_1,u_2\}$ and ${(\mathcal{H}·e_0)}^{*}={({\mathcal{H}}^{\mathcal{C}-C_0}·e_0)}^{*}=(X^{*},{\mathcal{C}}^{*},{\mathcal{D}}^{*})$. Then

$${\mathcal{C}}^{*}\subseteq \{C\in \mathcal{C}:C\cap e_0=\varnothing \}\cup \{(C-e_0)\cup \left\{w\right\}:C\in \mathcal{C},|C\cap e_0|=1\},$$

where $w\notin X$. As $\mathcal{H}$ is reduced, $C\in {\mathcal{C}}^{*}$ holds for each $C\in \mathcal{C}$ with $C\cap e_0=\varnothing$.

Assume that $(C_2-e_0)\cup \left\{w\right\}\notin {\mathcal{C}}^{*}$ for some $C_2\in \mathcal{C}$ with $|C_2\cap e_0|=1$. By the definition of ${\mathcal{H}}^{*}$, there exists $C_1\in \mathcal{C}-\{C_0,C_2\}$ with $C_1-e_0\subseteq C_2-e_0$. As $\mathcal{H}$ is reduced, $|C_1\cap e_0|=1$ and $C_1\cap e_0\ne C_2\cap e_0.$ However, Lemma 5 (i) implies that ${\mathcal{H}}^{\mathcal{C}-C_0}·e_0$ is uncolorable and so ${({\mathcal{H}}^{\mathcal{C}-C_0}·e_0)}^{*}$ is uncolorable, a contradiction. Thus,

$${\mathcal{C}}^{*}=\{C\in \mathcal{C}:C\cap e_0=\varnothing \}\cup \{(C-e_0)\cup \left\{w\right\}:C\in \mathcal{C},|C\cap e_0|=1\}.$$

As $e_0\subseteq C_0$ and $e_0\notin \mathcal{D}$, Lemma 1 (i) implies that $e_0⊈C$ for all $C\in \mathcal{C}-\left\{C_0\right\}$. Hence, $|{\mathcal{C}}^{*}|=|\mathcal{C}-\left\{C_0\right\}|=|\mathcal{C}|-1.$ □

Now we are ready to prove Theorem 4.

Proof of Theorem 4 

We shall prove Theorem 4 by showing that for any colorable and reduced mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$,

$${(-1)}^{\left|X\right|+\left|\mathcal{C}\right|}P(\mathcal{H},\lambda )>0$$

holds for all real $\lambda \in (-\infty ,0)$. We shall prove by induction on the size of $\left|\mathcal{C}\right|$.

In [14], it has been proved that for any reduced hypergraph $\mathcal{H}=(X,\mathcal{D})\in {\mathcal{L}}_0\subseteq {\mathcal{Q}}_0$, ${(-1)}^{\left|X\right|}P(\mathcal{H},\lambda )>0$ holds for all real $\lambda \in (-\infty ,0)$. Thus, the statement holds when $\left|\mathcal{C}\right|=0$.

Assume that the statement holds for any reduced and colorable mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ with $\left|\mathcal{C}\right|<k$$(k\ge 1)$. Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0$ be a reduced and colorable mixed hypergraph with $\left|\mathcal{C}\right|=k$ and $C_0=\{u_1,u_2,\dots ,u_p\}\in \mathcal{C}$.

Assume that $\{\{u_i,u_j\}\subseteq C_0:1\le i<j\le p,\{u_i,u_j\}\notin \mathcal{D}\}=\{e_1,e_2,\dots ,e_r\}$. Since $\mathcal{H}$ is colorable, we have $r\ge 1$. By Corollary 1 and Proposition 2, we have that

$$P(\mathcal{H},\lambda )=\sum _{i=0}^{r-1}P({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1},\lambda )=\sum _{i=0}^{r-1}P({({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*},\lambda ),$$

(14)

where ${\mathcal{H}}_0=\mathcal{H}$ and ${\mathcal{H}}_i={\mathcal{H}}_{i-1}{+}_d{e}_i$ for $i=1,2,\cdots ,r-1$. Since $\mathcal{H}\in {\mathcal{Q}}_0$ and $\mathcal{H}$ is reduced, for each $i=0,1,\dots ,r-1$, ${\mathcal{H}}_i\in {\mathcal{Q}}_0$ and ${\mathcal{H}}_i$ is reduced by Lemma 1 (ii). Thus, by Lemma 4, for each $i=0,1,\dots ,r-1$,

$${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}\in {\mathcal{Q}}_0.$$

Let I be the set of integers i with $0\le i\le r-1$ such that ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}$ is colorable. As $\mathcal{H}$ is colorable, (14) implies that $I\ne \varnothing$. For any $i\in I$, ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}=(X_i^{*},{\mathcal{C}}_i^{*},{\mathcal{D}}_i^{*})$ is colorable with $|X_i^{*}|=|X|-1$ and Lemma 5 implies that $|{\mathcal{C}}_i^{*}|=k-1$. By inductive assumption and (14), for any real $\lambda <0$,

$$\begin{array}{cc}\hfill {(-1)}^{\left|X\right|+k}P(\mathcal{H},\lambda )=& {(-1)}^{\left|X\right|+k}\sum _{i\in I}P({({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*},\lambda )\hfill \\ \hfill =& \sum _{i\in I}{(-1)}^{\left|X\right|+k-2}P({({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*},\lambda )>0.\hfill \end{array}$$

Hence, the statement holds and the proof is complete. □

For any colorable mixed hypergraph $\mathcal{H}$, $P(\mathcal{H},\lambda )$ is a polynomial in $\lambda$ (see [22]). By definition, $P(\mathcal{H},0)=0$. Thus,

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

(15)

is also a polynomial in $\lambda$. To prove Theorem 5, it suffices to establish the result as below.

Theorem 6.

Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0^{\prime }$. Assume that $\mathcal{H}$ is colorable and reduced. Then, for any real $\lambda \in [0,1)$,

$${(-1)}^{\left|X\right|+\left|\mathcal{C}\right|+1}Q(\mathcal{H},\lambda )>0.$$

(16)

Proof. 

We shall prove the statement holds by induction on the size of $\left|\mathcal{C}\right|$.

Furthermore, in [14], it has been shown that for any reduced hypergraph $\mathcal{H}=(X,\mathcal{D})\in {\mathcal{L}}_0^{\prime }\subseteq {\mathcal{Q}}_0^{\prime }$, ${(-1)}^{\left|X\right|+1}Q(\mathcal{H},\lambda )>0$ holds for all real $\lambda \in [0,1)$. Thus, the statement holds when $\left|\mathcal{C}\right|=0$.

Assume that the statement holds for any reduced and colorable mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0^{\prime }$ with $\left|\mathcal{C}\right|<k$$(k\ge 1)$. Let $\mathcal{H}=(X,\mathcal{C},\mathcal{D})\in {\mathcal{Q}}_0^{\prime }$ be a reduced and colorable mixed hypergraph with $\left|\mathcal{C}\right|=k$ and $C_0=\{u_1,u_2,\dots ,u_p\}\in \mathcal{C}$.

Assume that $\{\{u_i,u_j\}\subseteq C_0:1\le i<j\le p,\{u_i,u_j\}\notin \mathcal{D}\}=\{e_1,e_2,\dots ,e_r\}$. As $\mathcal{H}$ is colorable, $r\ge 1$. By Corollary 1 and (15),

$$Q(\mathcal{H},\lambda )=\sum _{i=0}^{r-1}Q({({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*},\lambda ),$$

(17)

where ${\mathcal{H}}_0=\mathcal{H}$ and ${\mathcal{H}}_i={\mathcal{H}}_{i-1}{+}_d{e}_i$ for $i=1,2,\cdots ,r-1$.

Let I be the set of integers i with $0\le i\le r-1$ such that ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}$ is colorable. As $\mathcal{H}$ is colorable, (17) implies that $I\ne \varnothing$.

For each $i\in I$, as $\mathcal{H}\in {\mathcal{Q}}_0^{\prime }$ and $e_{i+1}\subseteq C_0$ with $e_{i+1}\notin \mathcal{D}$, by Lemma 1 (ii), $|D\cap e_{i+1}|\le 1$ for each $D\in {\mathcal{D}}_2$, implying that ${\mathcal{H}}^2·e_{i+1}$ is connected, where ${\mathcal{H}}^2=(X,{\mathcal{D}}_2)$. Thus, ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}$ satisfies condition (IV). As $\mathcal{H}\in {\mathcal{Q}}_0^{\prime }\subset {\mathcal{Q}}_0$, by Lemma 4, ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}\in {\mathcal{Q}}_0^{\prime }$.

By Lemma 5, the number of $\mathcal{C}$-edges in ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}$ is exactly equal to $\left|\mathcal{C}\right|-1$. As $|e_{i+1}|=2$ for each $i\in I$, the order of ${({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*}$ is $\left|X\right|-1$. Thus, by induction assumption, for any real $\lambda \in [0,1)$,

$$\begin{array}{cc}\hfill {(-1)}^{\left|X\right|+\left|\mathcal{C}\right|+1}Q(\mathcal{H},\lambda )=& {(-1)}^{\left|X\right|+\left|\mathcal{C}\right|+1}\sum _{i\in I}Q({({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*},\lambda )\hfill \\ \hfill =& {(-1)}^2\sum _{i\in I}\underset{>0}{\underbrace{{(-1)}^{\left|X\right|+\left|\mathcal{C}\right|-1}Q({({\mathcal{H}}_i^{\mathcal{C}-C_0}·e_{i+1})}^{*},\lambda )}}>0.\hfill \end{array}$$

Hence, the statement holds and the proof is completed. □

6. Conclusions

Theorems 4 and 5 show that $(-\infty ,0)$ and $(0,1)$ are zero-free intervals for the chromatic polynomials of some families of mixed hypergraphs, which generalize the known results on zero-free intervals of chromatic polynomials of graphs and hyeprgraphs. However, we are not sure if they can be extended to larger families of hypergraphs and mixed hypergraphs. Especially for Theorem 5, we hope to find a sufficient condition, which is weaker than that in Theorem 5, for a mixed hypergraph $\mathcal{H}$ such that $P(\mathcal{H},\lambda )\ne 0$ for all real $\lambda \in (0,1)$. In addition, we wonder if there is a family of hypergraphs and mixed hypergraphs whose chromatic polynomials are zero-free in the interval $(1,32/27]$.