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
,
and
(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
. 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
and
.
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 consists of a vertex setX and two subsets and of the power set of X. The members in and are called -edges and -edges, respectively. A hypergraph is a mixed hypergraph with .
For any positive integer , a proper λ-coloring of a mixed hypergraph is an assignment of colors to vertices in such that each member in contains at least two vertices assigned the same color and each member in contains at least two vertices assigned different colors. Thus, the definition of proper -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 , containing one edge with .
Let be a mixed hypergraph and be a positive integer. The chromatic polynomial of , denoted by , is the function counting the number of proper -colorings of . Note that if 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 , if there exists such that for every pair of vertices , then . Next, we define an operation on mixed hypergraphs, which is based on the operation of identifying vertices in graphs. Let
be a mixed hypergraph and
. The mixed hypergraph
is obtained from
by removing all
with
, removing all
with
, and identifying all vertices in
as one vertex. Thus,
is the mixed hypergraph
, where
,
w is the new vertex when
is produced from
, and
For an edge , let denote the mixed hypergraph , where . We also say that is obtained from by contracting the edge E.
Clearly, for any colorable mixed hypergraph with a -edge C of size two, . Therefore, in the following, we need only to consider mixed hypergraphs satisfying for all .
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 -edge is at least 3. For any mixed hypergraph , we define the reduced subhypergraph to be the maximal reduced subhypergraph among all the subhypergraphs of . In other words, is obtained from by removing any edge or repeatedly whenever or holds for another edge or .
It is not difficult to show that is uniquely determined by , and if a mixed hypergraph is reduced, then . 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 be a mixed hypergraph and be the reduced subhypergraph of . Then . 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 be a colorable and reduced mixed hypergraph with . For any ,where . An alternative form of Theorem 1 was given by Voloshin [
27].
Theorem 2 ([
27])
. Let be a colorable and reduced mixed hypergraph. For any two vertices with , the chromatic polynomial of iswhere is a mixed hypergraph obtained from by adding a new -edge . 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 be a colorable and reduced mixed hypergraph with and let with . Suppose that . Then andwhere and for . Proof. As and for , we can write as for , where .
For each
, we apply Theorem 2 on
and obtain that
where
. Clearly,
contains a
-edge
in which every two-element subset
of
is a
-edge. Then by Proposition 1,
is uncolorable and so
for any positive integer
. 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 is defined to be a sequence of alternating vertices and edges where , are pairwise distinct vertices and whenever or for all , such that for all , where . Note that it is possible that for some . In such case, by the definition of a cycle, . A cycle is called a -cycle if for all .
Let be a mixed hypergraph and . A subset of X is said to be -independent if holds for every . Let . Sometimes, is written as for short. Then, is not -independent in if and only if holds for some .
Listed below are four conditions (I)–(IV) concerning a mixed hypergraph . Our main results of this paper are subject to some or all of them:
- (I)
is even for each ;
- (II)
For every -cycle of , the set is not -inde pendent in ;
- (III)
For every cycle of , which is not a -cycle, the set is not -independent in ; and
- (IV)
The spanning subhypergraph of is connected.
Let
be the set of hypergraphs satisfying conditions (I) and (II) and
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
and
.
Theorem 3 ([
14])
. For any hypergraph in , is zero-free in the interval ; Furthermore, if is in , then is zero-free in the interval and has no multiple zero at . In this article, we extend the above results to chromatic polynomials of mixed hypergraphs. Let be the set of mixed hypergraphs with satisfying conditions (I)–(III) and be the set of mixed hypergraphs with satisfying conditions (I)–(IV). Clearly, and . We prove the following two results on the chromatic polynomials of mixed hypergraphs in and .
Theorem 4. For any colorable mixed hypergraph in , is zero-free in the interval .
Observe that in
Figure 1 the hypergraph
does not belong to
, while the hypergraph
belongs to
. By applying Corollary 1 and Theorem 1 repeatedly, it can be calculated that
which has a real zero at
, and
which shows that
has no real zeros in the interval
.
Theorem 5. For any colorable mixed hypergraph in , is zero-free in the interval . Furthermore, has no multiple zero at .
Theorems 4 and 5 imply Theorem 3 directly, as
and
. The proofs of Theorems 4 and 5 are given in
Section 5.
4. Is Deletion and Contraction Closable
In this section, we show that is deletion and contraction closable; i.e., for any mixed hypergraph with , at least one of the following cases happens:
- (1)
There exists some such that both and hold;
- (2)
There exists some such that both and hold, where with and .
Let us first study the structure of a mixed hypergraph which satisfies condition (III).
Lemma 1. Let be a reduced mixed hypergraph satisfying condition (III). Then,
- (i)
For any distinct pair with , holds;
- (ii)
For any and with , holds; and
- (iii)
If is colorable, then .
Proof. - (i)
Suppose that there exist two vertices with . By definition, forms a cycle of . Evidently, is not a -cycle of . As satisfies condition (III), , a contradiction.
- (ii)
Assume that there exist and with such that Let , where and . By definition, forms a cycle, but not a -cycle, of . Furthermore, as satisfies condition (III), . As with , , implying that is not reduced, a contradiction.
- (iii)
Assume that there exists . By the definition of a proper coloring of a mixed hypergraph, if is colorable and , then . From the result (ii) above, we have , a contradiction. □
For any colorable and reduced mixed hypergraph , by Lemma 1 (iii), holds.
Let be a reduced mixed hypergraph and with . It is possible that is not reduced. By Proposition 2, we consider instead of in this paper. As when , in what follows, we write both and as and by the notation for the reduced mixed subhypergraphs, both and are written as . Hence, .
Lemma 2. Let be a reduced mixed hypergraph and with . If with , then .
Proof. By the definition of
,
where
. Suppose that there exists
with
but
. By the definition of the reduced mixed subhypergraph, there must be an edge
such that
, implying
as
. As
is reduced,
. By (
6), there is an edge
such that
. As
,
in
. However, if
, by (
6) again, then
, implying that
, a contradiction. □
Lemma 3. If is a reduced mixed hypergraph satisfying condition (II), then for any with ,holds, where . Proof. By the definition of
, we have that
Thus, it suffices to show that for any with , there exists with such that .
Let with . Observe that forms a -cycle in . As satisfies condition (II), there exist two distinct vertices such that . As is a reduced mixed hypergraph, for both , implying, without loss of generality, that and . Thus, . □
Now we are ready to show that
is deletion and contraction closable. In [
14], the authors of this article proved that for any reduced mixed hypergraph
with
,
and
; i.e.,
is deletion and contraction closable. When
with
, we shall show that
is also deletion and contraction closable by completing the following lemma.
Lemma 4. Let with and with . Assume that is a reduced mixed hypergraph and . Then, and .
Proof. Since and , it is clear that . To show that is equivalent to showing that satisfies conditions (I)–(III).
By the definition of
, we have that
As
,
and
, by Lemma 1 (ii),
holds for all
. Thus, (
9) can be rewritten as below:
By the definition of , , implying that satisfies condition (I) as .
By definition,
where
, and
If contains no cycles, by Claim 1, . Now we assume that contains cycles. Let be any cycle in , where and for each . We shall complete the proof of Claim 2 by the following two sub-claims:
Claim 2.1: If
is a
-cycle, then there exists an edge
such that
From (
11), there exist edges
in
with the following properties: for all
,
- (a)
if and only if ;
- (b)
;
- (c)
when and otherwise.
Now we are going to prove Claims 2.1 and 2.2 on a case-by-case basis.
In this case, is a cycle in .
If
is a
-cycle in
, then
for all
, implying that
for all
. Thus,
is a
-cycle in
and there is an edge
such that
. As
, by Lemma 1 (ii),
. If
, by (
11) and Lemma 2,
and
. If
, then either
or
. Without loss of generality, we assume that
. Thus,
and
. By (
11) and Lemma 2, there is an edge
where
. Thus, Claim 2.1 holds in this case.
If
is not a
-cycle in
, then there exists
for some
, implying that
. Thus,
is not a
-cycle in
and there is an edge
for some
. As
in this subcase,
. By (
11) and Lemma 2,
. Thus, Claim 2.2 holds in this case.
Without loss of generality, suppose that
. As
, by (
11), we have
, while
for all
.
Clearly, is a cycle in . Thus, it can be proved similarly as in Case 1 that Claims 2.1 and 2.2 hold in this subcase.
By (b),
for all
. As
,
and
,
is a cycle, but not a
-cycle, in
as
. As
, there is an edge
such that
. Furthermore, as
,
, which implies that
.
If
, by (
11) and Lemma 2, then
and
. Thus, Claims 2.1 and 2.2 hold.
If
, then either
or
. Without loss of generality, we assume that
and
for some
. Thus,
and
. By (
11) and Lemma 2, there is an edge
As in this case, , 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 with . Assume that is reduced. For any and with ,
- (i)
If there exist distinct -edges with for , and , then is uncolorable, where ;
- (ii)
If is colorable, thenwhere . Moreover, .
Proof. - (i)
Let
and
. Thus,
As , by Lemma 1 (i), for each pair . Moreover, is a cycle in , where v is any vertex in . As and , for each , holds for some .
Let
. As
, by Lemma 1 (ii),
for each
. By the definition of
,
, where
, and
As
is reduced,
and
,
As for any
, either
or
, by (
13),
for each
. As
for each pair
and
, by (
13),
for each pair
. Thus, by Proposition 1,
is uncolorable.
- (ii)
Let
and
. Then
where
. As
is reduced,
holds for each
with
.
Assume that
for some
with
. By the definition of
, there exists
with
. As
is reduced,
and
However, Lemma 5 (i) implies that
is uncolorable and so
is uncolorable, a contradiction. Thus,
As and , Lemma 1 (i) implies that for all . Hence, □
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
,
holds for all real
. We shall prove by induction on the size of
.
In [
14], it has been proved that for any reduced hypergraph
,
holds for all real
. Thus, the statement holds when
.
Assume that the statement holds for any reduced and colorable mixed hypergraph with . Let be a reduced and colorable mixed hypergraph with and .
Assume that
. Since
is colorable, we have
. By Corollary 1 and Proposition 2, we have that
where
and
for
. Since
and
is reduced, for each
,
and
is reduced by Lemma 1 (ii). Thus, by Lemma 4, for each
,
Let
I be the set of integers
i with
such that
is colorable. As
is colorable, (
14) implies that
. For any
,
is colorable with
and Lemma 5 implies that
. By inductive assumption and (
14), for any real
,
Hence, the statement holds and the proof is complete. □
For any colorable mixed hypergraph
,
is a polynomial in
(see [
22]). By definition,
. Thus,
is also a polynomial in
. To prove Theorem 5, it suffices to establish the result as below.
Theorem 6. Let . Assume that is colorable and reduced. Then, for any real , Proof. We shall prove the statement holds by induction on the size of .
Furthermore, in [
14], it has been shown that for any reduced hypergraph
,
holds for all real
. Thus, the statement holds when
.
Assume that the statement holds for any reduced and colorable mixed hypergraph with . Let be a reduced and colorable mixed hypergraph with and .
Assume that
. As
is colorable,
. By Corollary 1 and (
15),
where
and
for
.
Let
I be the set of integers
i with
such that
is colorable. As
is colorable, (
17) implies that
.
For each , as and with , by Lemma 1 (ii), for each , implying that is connected, where . Thus, satisfies condition (IV). As , by Lemma 4, .
By Lemma 5, the number of
-edges in
is exactly equal to
. As
for each
, the order of
is
. Thus, by induction assumption, for any real
,
Hence, the statement holds and the proof is completed. □