Some Results on Chromatic Polynomials of Hypergraphs

Abstract

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/Łazuka, Dohmen and Tomescu. Furthermore, it is shown that the class of (non-uniform) hypertrees with $m$ edges, where $m_r$ edges have size $r$, $r\geq 2$, is chromatically closed if and only if $m\leq4$, $m_2\geq m-1$.

Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs

“…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]).…”

“…It may have been noticed to be a polynomial by Chvátal [11]. It has been studied extensively in the past twenty years by many researchers, such as Allagan [1,2,3], Borowiecki and Lazuka [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.…”

Problems on chromatic polynomials of hypergraphs

“…The problem of determining the family H = {H : P (H, λ) = P (H , λ)} for a given hypergraph H is very difficult. Some families H have been confirmed in [5,6,19,20,21,23,24]. A hypergraph H = (V, E) is said to be h-uniform if |e| = h holds for all e ∈ E. It is unknown if all hypergraphs in H are h-uniform for any given h-uniform hypergraph H.…”

“…The graph-function P (H, λ) for a hypergraph H appeared in the work of Helgason [15] in 1972, but it is unknown if it had been mentioned earlier. It has been studied extensively in the past twenty years by many researches, such as Allagan [1,2,3], Borowiecki and Łazuka [5,6], Dohmen [8,9], Drgas-Burchardt and Łazuka [13], Tomescu [19,20,21,22,23] and Walter [24]. A number of properties of chromatic polynomials of graphs related to their expressions, computations and factorizations, etc, have been extended to chromatic polynomials of hypergraphs.…”

Some Results on Chromaticity of Quasi-Linear Paths and Cycles

“…where Q(λ) is a polynomial of degree at most equal to n − 3h + r + 2, α 1 = m − 1 is the number of subpaths P h,r 2 of length two, α 2 = m 2 − m + 1 is the number of pairs of pairwise disjoint edges and α 3 = m − 2 is the number of subpaths P h,r 3 of length three in P h,r m . Also since any spanning subhypergraph of P h,r m induced by less than m edges is not connected, it follows that in (5) the coefficient of λ is (−1) m , which implies that H is also connected [15]. Since H has all edges of cardinality h, it follows that the number of components of a spanning subhypergraph of H may be n, n − h + 1 or a smaller number.…”