Chromatic polynomials of hypergraphs

Let H = (X, E) be a simple hypergraph and let f (H, ) denote its chromatic polynomial. Two hypergraphs H 1 and H 2 are chromatic equivalent if f (H 1 , ) = f (H 2 , ). The equivalence class of H is denoted by H . Let K and H be two classes of hypergraphs. H is said to be chromatically characterized in K if for every HIn this paper we prove that uniform hypertrees and uniform unicyclic hypergraphs are chromatically characterized in the class of linear hypergraphs.

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“…Theorem 2 (Borowieiki and Lazuka [2]). If H is a hypergraph such that H = H 1 ∪ · · · ∪ H k for k 2, where…”

Section: Some Known Resultsmentioning

confidence: 94%

“…Afterwards many scientists, among them Dohmen, Jones and Tomescu [4][5][6]8], started to study the chromaticity of hypergraphs. Till now only few chromatically equivalent or chromatically unique hypergraphs are known [2,8].…”

Section: Introductionmentioning

confidence: 99%

On chromaticity of hypergraphs

Borowiecki

Łazuka

2007

Discrete Mathematics

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“…Note that this terminology goes back to Erdös and Rado [5]. [2]). SH(n, 1, h) is chromatically unique.…”

Section: Notation and Preliminary Resultsmentioning

confidence: 95%

“…Note that for p = h − 1, SH(n, h − 1, h) is an h-uniform hypertree and its chromatic polynomial (3) coincides with the expression given by (2).…”

Section: H-chromatic Uniqueness Of Sh(n P H)mentioning

confidence: 96%

On the chromaticity of sunflower hypergraphs SH(n,p,h)

Tomescu

2007

Discrete Mathematics

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A sunflower hypergraph SH (n, p, h) is an h-hypergraph of order n = h + (k − 1)p and size k (1 p h − 1 and h 3), where each edge (or a "petal") consists of p distinct vertices and a common subset to all edges with h − p vertices. In this paper, it is shown that this hypergraph is h-chromatically unique (i.e., chromatically unique in the set of all h-hypergraphs) for every 1 p h − 2, but this is not true for p = h − 1 and k 3. Also SH(n, p, h) is not chromatically unique for every p, k 2. Notation and preliminary resultsA simple hypergraph H = (V , E), with order n = |V | and size m = |E|, consists of a vertex-set V (H ) = V and an edge-set E(H ) = E, where E ⊆ V and |E| 2 for each edge E in E. H is h-uniform, or is an h-hypergraph, if |E| = h for each E in E and H is linear if no two edges intersect in more than one vertex. A hypergraph, for which no edge is a subset of any other is called Sperner. Two vertices u, v

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“…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.…”

Section: The Electronic Journal Of Combinatorics 16 (2009) #R94mentioning

confidence: 99%

Some Results on Chromatic Polynomials of Hypergraphs

Walter¹

2009

Electron. J. Combin.

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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 $m\leq4$, $m_2\geq m-1$.

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Chromatic polynomials of hypergraphs

Cited by 15 publications

References 4 publications

On chromaticity of hypergraphs

On chromaticity of hypergraphs

On the chromaticity of sunflower hypergraphs SH(n,p,h)

Some Results on Chromatic Polynomials of Hypergraphs

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