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Properties of Chromatic Polynomials of Hypergraphs Not Held for Chromatic Polynomials of Graphs∗

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Properties of Chromatic Polynomials of Hypergraphs Not Held for Chromatic Polynomials of Graphs∗ Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs∗ Ruixue Zhang, Fengming Dongy Mathematics and Mathematics Education National Institute of Education Nanyang Technological University, Singapore 637616 Abstract In this paper, we present some properties on chromatic polynomials of hy- pergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph G = (V; E), the number of totally cyclic orientations of G is equal to the value of jP (HG; −1)j, where P (HG; λ) is the chromatic polyno- mial of a hypergraph HG which is constructed from G. Finally we show that the multiplicity of root \0" of P (H; λ) may be at least 2 for some connected hypergraphs H, and the multiplicity of root \1" of P (H; λ) may be 1 for some connected and separable hypergraphs H and may be 2 for some connected and non-separable hypergraphs H. 1 Introduction and main results For any graph G = (V; E), the chromatic polynomial of G is the function P (G; λ) arXiv:1611.04245v3 [math.CO] 21 Apr 2017 such that for any positive integer λ, P (G; λ) is the number of proper λ-colourings of G, where a proper λ-colouring of G is a mapping φ : V ! f1; 2; ··· ; λg such that φ(u) 6= φ(v) holds for each pair of adjacent vertices u and v in G. This graph-function P (G; λ) was originally introduced by Birkhoff [6] in 1912 in the hope of proving the four-color theorem (i.e., P (G; 4) > 0 holds for any loopless planar graph G). A hypergraph H consists of an order pair of vertex set V and edge set E, where E is a subset of fe ⊆ V : jej ≥ 1g. If jej ≤ 2 for all e 2 E, then H is a graph. ∗This paper was partially supported by NTU AcRf Project RP 3/16 DFM of Singapore. yCorresponding author. Email: [email protected] 1 For any integer λ ≥ 1, a weak proper λ-colouring of a hypergraph H = (V; E) is a mapping φ : V ! f1; 2; ··· ; λg such that jfφ(v): v 2 egj > 1 holds for each e 2 E (see [25, 26, 41]). Thus H does not have any weak proper λ-colouring if jej = 1 for some edge e 2 E.A strong proper λ-colouring of H = (V; E) is a mapping φ : V ! f1; 2; ··· ; λg such that jfφ(v): v 2 egj = jej holds for each e 2 E (see [25, 26, 41] also). Note that the number of distinct strong proper λ-colourings in H = (V; E) is equal to P (G; λ), where G is the simple graph with vertex set V in which any two distinct vertices u; v 2 V are adjacent if and only if they are contained in an edge e 2 E in H (i.e., G is obtained from H by changing each edge in H to a clique in G). Thus the function counting the number of strong proper λ-colourings in H is not different from the chromatic polynomial of a graph. It is probably for this reason that most articles on chromatic polynomials of hypergraphs in the past two decades focused on the function counting the number of weak proper λ-colourings in a hypergraph H (see [1, 2, 3, 8, 13, 36, 37, 38, 39, 41, 42]). Let P (H; λ) be the number of weak proper λ-colourings of H. It is obvious that this graph-function P (H; λ) is an extension of the chromatic polynomial of a graph. In this paper, P (H; λ) is called the chromatic polynomial of H, and it is indeed a polynomial in λ of degree jVj. This graph-function P (H; λ) appeared in the work of Helgason [20] in 1972, and it is unknown if it had been introduced earlier. It may have been noticed to be a polynomial by Chv´atal[11]. It has been studied extensively in the past twenty years by many researchers, such as Allagan [1, 2, 3], Borowiecki andLazuka [8], Dohmen [13], Tomescu [36, 37, 38, 39], Voloshin [41] and Walter [42]. They extended many properties of chromatic polynomials of graphs on computations, expressions, factorizations, etc, to chromatic polynomials of hypergraphs. The following are some known properties on chromatic polynomials of graphs which also hold for chromatic polynomials of hypergraphs: (A.1). Deletion/Contraction property [29, 30, 25, 45], cited as Theorem 5; (A.2). (by definition) multiplicativity with respect to disjoint unions, cited as Propo- sition 2; (A.3). the factorization formula for the chromatic polynomial of a graph with a cut-set which induced a clique [8, 25, 47], cited as Theorem 6; (A.4). the roots of chromatic polynomials of graphs are dense in the complex plane [32]; (A.5). (as the family of hypergraphs include all graphs) the real roots of chromatic polynomials of graphs are dense in the interval [32=27; 1) (see [23, 35]); 2 (A.6). Whitney's Broken-cycle Theorem [45], extended by Dohmen [13] to the set of linear hypergraphs which do not contain any edge with an odd size and contain edges of size 2 in every cycle; (A.7). for fixed λ, computing P (H; λ) is polynomial time computable for classes of graphs (hypergraphs) of bounded tree width. The proof also works for hyper- graphs [27]; n n−1 (A.8). for any graph G of order n, we have P (G; λ) = λ + an−1λ + ··· + a1λ, where P j ai = j≥0(−1) N(i; j) and N(i; j) denotes the number of spanning subgraphs of G with i components and j edges [5, 7, 14, 29, 30, 45]. Tomescu [36] showed that this result also holds chromatic polynomials of hypergraphs (see [13] also); (A.9). 0 is a root of every chromatic polynomial and all positive integers are roots of chromatic polynomials [5, 7, 13, 14, 29, 30, 36, 45]. But chromatic polynomials of graphs also have the following properties on its coeffi- cients not held for chromatic polynomials of hypergraphs: (B.1). for any graph G of order n and component number c, if P (G; λ) = λn + n−1 an−1λ +···+a1λ, then ai 6= 0 if and only if c ≤ i ≤ n (see [5, 7, 14, 29, 30, 45]). Instead, for a hypergraph H = (V; E) of order n with jej ≥ 3 for all e 2 E, (A.8) n n−1 implies that if P (H; λ) = λ + bn−1λ + ··· + b1λ, then bn−i 6= 0 but bn−j = 0 for all j with 1 ≤ j < i, where i = mine2E jej − 1 (see [13, 36]); (B.2). the coefficients 1; an−1; ··· ; ac in the expansion of P (G; λ) alternate in signs (see [5, 7, 14, 29, 30, 45]). Instead, for any linear hypergraph H = (V; E) (i.e., je1 \ e2j ≤ 1 for all distinct e1; e2 2 E) with jVj = n and h = mine2E jej ≥ 3, if n n−1 H contains edges of size h + 1 and P (H; λ) = λ + bn−1λ + ··· + b1λ, then (A.8) implies that bn−h+1 and bn−h have the same sign (see [13, 36]); 2 (B.3). the sequence 1; jan−1j; ··· ; ja2j; ja1j is log-concave (i.e., aj ≥ jaj−1j · jaj+1j holds for all j), where ai's are coefficients of terms of P (G; λ) given in (B.1) (see n n−1 [21, 22, 29, 30]). Instead, for a hypergraph H, if P (H; λ) = λ + bn−1λ + ··· + b1λ, by (B.1), it is possible that bn−j = 0 while bn−k 6= 0 and bn−i 6= 0 for some integers k; j; i with 0 ≤ k < j < i (see [13, 36]). In this article, we will present some properties on chromatic polynomials of hyper- graphs which are different from the following properties on chromatic polynomials of graphs: (i).( −∞; 0), (0; 1) and (1; 32=27] are zero-free intervals for chromatic polynomials of graphs [23, 30, 35], while chromatic polynomials of hypergraphs have dense real zeros in the whole set of real numbers, as stated in Corollary 1 (a); 3 (ii). for any graph G, jP (G; −1)j counts the number of acyclic orientations (i.e., orientations without any directed cycle) of a multigraph G (see [33]), while jP (HG; −1)j counts the number of totally cyclic orientations of G (i.e., orienta- tions in which each arc is contained in some directed cycle), as stated in Corol- lary 2, where HG is obtained from G by adding jEj new vertices fwe : e 2 Eg and changing each edge e in G to an edge fue; ve; weg in HG, where ue and ve are the two ends of e in G; (iii). the multiplicity of root `0' of P (G; λ) counts the number of components of a graph G, and thus P (G; λ) has no factor λ2 whenever G is connected (see [14, 16, 29, 30]), while P (H; λ) may have a factor λ2 for a connected hypergraph H, as stated in Theorem 3; (iv). for a connected graph G, P (G; λ) has a factor (λ−1)2 if and only if G is separable (see [44, 46]), while P (H; λ) may have no factor (λ − 1)2 for a connected and separable hypergraph H, as stated in Theorem 4.
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