Some Results on Chromatic Polynomials of Hypergraphs

In this paper, chromatic polynomials of (non-uniform) hypercycles, unicyclic hypergraphs, hypercacti and sunflower hypergraphs are presented. The formulae generalize known results for r-uniform hypergraphs due to Allagan, Borowiecki/ Lazuka, Dohmen and Tomescu. Furthermore, it is shown that the class of (non-uniform) hypertrees with m edges, where mr edges have size r, r ≥ 2, is chromatically closed if and only if m ≤ 4, m2 ≥ m− 1. 1 Notation and preliminaries Most of the notation concerning graphs and hypergraphs is based on Berge [4]. A hypergraph H = (V , E) consists of a finite non-empty set V of vertices and a family E of edges which are non-empty subsets of V of cardinality at least 2. An edge e of cardinality r(e) is called an r-edge. H is r-uniform if each edge e ∈ E is an r-edge. The degree dH(v) is the number of edges containing the vertex v. A vertex v is called pendant if dH(v) = 1. H is said to be simple if all edges are distinct. H is is said to be Sperner if no edge is a subset of another edge. Uniform simple hypergraphs are Sperner. Simple 2-uniform hypergraphs are graphs. A hypergraph H′ = (W ,F) with W ⊆ V and F ⊆ E is called a subhypergraph of H. If W = ⋃ e∈F e, then the subhypergraph is said to be induced by F , abbreviated by HF . The 2-section of a hypergraph H = (V , E) is the graph [H]2 = (V , [E ]2) such that {u, v} ∈ [E ]2 , u 6= v, u, v ∈ V if and only if u, v are contained in a hyperedge of H. In a hypergraph H = (V , E) an alternating sequence v1, e1, v2, e2, . . . , em, vm+1, where vi 6= vj, 1 ≤ i < j < m, vi, vi+1 ∈ ei is called a chain. Note that repeated edges are the electronic journal of combinatorics 16 (2009), #R94 1 allowed in a chain. If also ei 6= ej, 1 ≤ i < j ≤ m, we call it a path of length m. If v1 = vm+1, a chain is called cyclic chain, and a path is called cycle. The subhypergraph C induced by the edge set of a cycle of length m is called a hypercycle, short m-hypercycle. Observe that in case of graphs the notion chain and path, cyclic chain and cycle coincide whereas this is not the case for hypergraphs in general. A hypergraph H is said to be connected if for every v, w ∈ V there exists a sequence of edges e1, . . . , ek, k ≥ 1 such that v ∈ e1, w ∈ ek and ei ∩ ei+1 6= ∅, for 1 ≤ i < k. The maximal subhypergraphs which are connected are called components. If a single vertex v or single edge e is a component then v or e is called isolated. We use the abbreviation ∪· for the disjoint union operation, especially of connected components. According Acharya [1], the relation ∼ in E is an equivalence relation, where e1 ∼ e2 if and only if e1 = e2 or there exists a cyclic chain containing e1, e2. A block of H is either an isolated vertex/edge or a subhypergraph induced by the edge set of an equivalence class. A block consisting of only one non-isolated edge is called a bridge-block. Lemma 1.1 ( [1, Theorem 1.1]). Two distinct blocks of a hypergraph have at most one vertex in common. The block-graph bc(H) of a hypergraph H = (V , E) is the bipartite graph created as follows. Take as vertices the blocks of H and the vertices in V which are common vertices of two blocks. Two vertices of bc(H) are adjacent if and only if one vertex corresponds to a block B of H and the other vertex is a common vertex c ∈ B. Observe that in case of graphs we get the block-cutpoint-tree introduced by Harary and Prins [10]. Lemma 1.2 ( [10, Theorem 1]). If G is a connected graph, then bc(G) is a tree A hypercycle C is said to be elementary if dC(vi) = 2 for each i ∈ {1, 2, . . . ,m} and each other vertex u ∈ ⋃m i=1 ei is pendant. This is equivalent to the fact that C contains only a unique cycle (sequence) up to permutation. A 2-uniform m-hypercycle (which is elementary per se) is called m-gon. A hypergraph is linear if any two of its edges do not intersect in more than one vertex. Elementary 2-hypercycles are not linear, whereas elementary m-hypercycles, m ≥ 3, are linear. A hypertree is a connected hypergraph without cycles. Obviously, a hypertree is linear. A hyperstar is a hypertree where all edges intersect in one vertex. A hyperforest consists of components each of which is a hypertree. A unicyclic hypergraph is a connected hypergraph containing exactly one cycle, i.e. one hypercycle which is elementary. A hypercactus is a connected hypergraph, where each block is an elementary hypercycle or a bridge-block. Note that this is another approach to generalize the notion of cactus from graphs to hypergraphs as chosen by Sonntag [14,15]. A hypergraph H = (V , E) of order n is called a sunflower hypergraph if there exist X ⊂ V , |X | = q, 1 ≤ q < n and a partition V \X = ⋃ · i=1Yi such that E = ⋃m i=1(X ∪· Yi). Each set Yi is called a petal, the vertices in X are called seeds. Observe, if |X | = 1 then H is a hyperstar and if |X | = 2 then H is a 2-hypercycle. the electronic journal of combinatorics 16 (2009), #R94 2 A λ-coloring of H is a function f : V → {1, . . . , λ}, λ ∈ N, such that for each edge e ∈ E there exist u, v ∈ e, u 6= v, f(u) 6= f(v). The number of λ-colorings of H is given by a polynomial P(H, λ) of degree n in λ, called the chromatic polynomial of H. Two hypergraphs H and H′ are said to be chromatically equivalent, written H ≈ H′, if and only if P(H, λ)=P(H′, λ). The equivalence class of H is abbreviated by 〈H〉. Extending a definition based on Dong, Koh and Teo [8, Chapter 3] from graphs to hypergraphs, a class H of hypergraphs is called chromatically closed if for any H ∈ H the condition 〈H〉 ⊆ H is satisfied. Let H,K be two classes of hypergraphs, then H is said to be chromatically closed within the class K, if for every H ∈ H ∩ K we have 〈H〉 ∩K ⊆ H ∩K. We use the following abbreviations throughout this paper. If H is isomorphic to H′, we write H ∼= H′. If H = H1 ∪ H2, H1 ∩ H2 ∼= Kn, we write H = H1 ∪n H2. Kn denotes the complete graph of order n, especially K1 is an isolated vertex. Kn denotes the hypergraph consisting of n ≥ 2 isolated vertices. S(k1)r1,...,(km)rm denotes a hyperstar with ki ri-edges, i = 1, . . . ,m. Cr1,...,rm denotes the elementary m-hypercycle, where ei has size ri, i = 1, . . . ,m. If ki consecutive edges of the hypercycle have the same size ri, we write C(k1)r1,...,(km)rm . Explicit expressions of chromatic polynomials of hypergraphs were obtained by several authors. In most cases the hypergraphs are assumed to be uniform and linear. The chromatic polynomials of r-uniform hyperforests and r-uniform elementary hypercycles were presented by Dohmen [7] and rediscovered by Allagan [3] who used a slightly different notation. Theorem 1.1 ( [7, Theorem 1.3.2, Theorem 1.3.4], [3, Theorem 1, Theorem 2]). If H = (V , E) is an r-uniform hyperforest with m edges and c components, where r ≥ 2, then P (H, λ) = λc(λr−1 − 1) (1.1) If H = (V , E) is an r-uniform elementary m-hypercycle, where r ≥ 2, m ≥ 3, then P (H, λ) = (λr−1 − 1) + (−1)(λ− 1) (1.2) With the restriction that the hypergraphs are linear, Borowiecki/ Lazuka [6] were able to show the converse of (1.1). Combined with the classical result of Read [13] concerning trees, we get Theorem 1.2 ( [6, Theorem 5], [13, Theorem 13]). If H is a linear hypergraph and P (H, λ) = λ(λr−1 − 1), where r ≥ 2,m ≥ 1 (1.3) then H is an r-uniform hypertree with m edges. Similarly, results of Eisenberg [9], Lazuka [12] for graphs and Borowiecki/ Lazuka [6] concerning r-uniform unicyclic hypergraphs, r ≥ 3, can be summarized as follows: the electronic journal of combinatorics 16 (2009), #R94 3 Theorem 1.3 ( [9], [12, Theorem 2], [6, Theorem 8]). Let H be a linear hypergraph. H is an r-uniform unicyclic hypergraph with m+ p edges and a cycle of length p if and only if P (H, λ) = (λr−1 − 1) + (−1)(λ− 1)(λr−1 − 1), (1.4) where r ≥ 2, m ≥ 0 and p ≥ 3. In parallel Allagan [3, Corollary 3] discovered a slightly different formula for r-uniform unicyclic hypergraphs which can be easily transformed into (1.4). Borowiecki/ Lazuka [5, Theorem 5] were the first who studied a class of non-linear uniform hypergraphs which are named sunflower hypergraphs by Tomescu in [17]. In [18] Tomescu gave the following formula of the corresponding chromatic polynomial which we restate in a slightly different notation. Theorem 1.4 ( [18, Lemma 2.1]). Let S(m, q, r) be an r-uniform sunflower hypergraph having m petals and q seeds, where m ≥ 1, 1 ≤ q ≤ r − 1, then P (S(m, q, r), λ) = λ(λr−q − 1) + λ(r−q)m(λq − λ) (1.5) The first formulae of chromatic polynomials of non-uniform hypergraphs were mentioned by Allagan [2]. He considered the special case of non-uniform elementary cycles Hm which are constructed from an m-gon, m ≥ 3, by replacing a 2-edge by a k+-edge, where k ≥ 1. Theorem 1.5 ( [2, Theorem 1]). The chromatic polynomial of the hypergraph Hm, m ≥ 3, has the form: P (Hm, λ) = (λ− 1) k ∑ i=0 λ + (−1)(λ− 1). (1.6) Remark 1.1. (1.6) can be restated as follows P (Hm, λ) = (λ− 1)m−1(λk+1 − 1) + (−1)(λ− 1) (1.7) Borowiecki/ Lazuka [5] extended (1.1) by dropping the uniformity assumption. Theorem 1.6 ( [5, Theorem 8]). If H = (V , E) is a hyperforest with mr r-edges, where 2 ≤ r ≤ R, and c components, then P (H, λ) = λ R ∏ r=2 (λr−1 − 1)r (1.8) These results suggest to generalize (1.2), (1.4) and (1.5) to non-uniform hypergraphs. Before we state our results, we remember three useful reduction methods concerning the calculation of chromatic polynomials of hypergraphs. Given a hypergraph H. If dropping an edge e ∈ E yields a hypergraph H′ being chromatically equivalent to H, then e is called chromatically inactive. Otherwise, e is said to be chromatically active. Dohmen [7] gave the following lemma: the electronic journal of combinatorics 16 (2009), #R94 4 Lemma 1.3 ( [7, Theorem 1.2.1]). A hypergraph H and the subhypergraph H′ which results by dropping all chromatically inactive edges are chromatically equivalent. The next lemma generalizes Whitney’s fundamental reduction theorem. It was already mentioned by Jones [11] in case where the added edge is a 2-edge. Lemma 1.4. Let H = (V , E) be a hypergraph, X ⊆ V an r-set, r ≥ 2, such that e * X for every e ∈ E. Let H+X denote the hypergraph obtained by adding X as a new edge to E and dropping all chromatically inactive edges. Let H.X be the hypergraph obtained by contracting all vertices in X to a common vertex x and dropping all chromatically inactive edges. Then P (H, λ) = P (H+X,λ) + P (H.X, λ) (1.9) Proof. We extend the standard proof well-known in the case of graphs. Let f be a λ-coloring of H and X ⊆ V an r-set, r ≥ 2, such that e * X for every e ∈ E . Either (i) there exist u, v ∈ X with f(u) 6= f(v) or (ii) f(u) = f(v) for all u, v ∈ X. The λ-colorings of H for which (i) holds are also λ-colorings of H+X = (V , E+X) where E+X = E ∪X \ EX where EX = {e ∈ E | X ⊂ e}, and vice versa. The λ-colorings of H for which (ii) holds are also λ-colorings of H.X = (V .X, E .X) where V .X = V \ X ∪ {x} , E .X = {e \X ∪ {x} | e ∈ E}, and vice versa. Observe that H.X may contain parallel edges, of which all but one can be dropped as chromatically inactive edges. Corollary 1.1. Let H = (V , E) be a hypergraph. Let H−e denote the hypergraph obtained by deleting some e ∈ E and let H.e be the hypergraph by contracting all vertices in e to a common vertex x and dropping all chromatically inactive edges. Then P (H, λ) = P (H−e, λ)− P (H.e, λ) (1.10) Borowiecki/ Lazuka [5] generalized an old result of Read [13]. Lemma 1.5 ( [5, Theorem 6]). If H is a hypergraph such that H = ⋃k i=1Hi for k ≥ 2, where Hi ∩Hj = Kp for i 6= j and ⋂k i=1Hi = Kp, then P (H, λ) = P (Kp, λ)1−k k ∏ i=1 P (Hi, λ). (1.11) 2 The chromatic polynomials of non-uniform hypergraphs Our first generalization concerns non-uniform elementary hypercycles. Note, that elementary 2-hypercycles are not linear whereas elementary m-hypercycles, m ≥ 3, are linear. the electronic journal of combinatorics 16 (2009), #R94 5 Theorem 2.1. If C = (V , E) is an elementary m-hypercycle having mr r-edges, where 2 ≤ r ≤ R, then