AN INVARIANT FOR HYPERGRAPHS

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Enumeration of Maximum Acyclic Hypergraphs

Abstract Acyclic hypergraphs are analogues of forests in graphs.They are very useful in the design ofdatabases. In this article,the maximum size of an acvclic hypergraph is determined and the number of maximumγ-uniform acyclic hypergraphs of order n is shown to be (_(r-1)~n)(n(r-1)-r~2 2r)~(n-r-1).     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

Optimal acyclic edge‐coloring of cubic graphs

An acyclic edge‐coloring of a graph is a proper edge‐coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge‐coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K₄ or K_3,3; the acyclic chromatic index of K₄ and K_3,3 is 5. This result has previously been published by Fiam?ík, but his published proof was erroneous.     

Acyclic edge coloring of graphs with maximum degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a′(G)?Δ + 2, where Δ=Δ(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Δ(G)?4, with the additional restriction that m?2n?1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m?2n, when Δ(G)?4. It follows that for any graph G if Δ(G)?4, then a′(G)?7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 192–209, 2009     

Acyclic Hypergraph Projections

Hypergraphs can be partitioned into two classes: tree (acyclic) hypergraphs and cyclic hypergraphs. This paper analyzes a new class of cyclic hypergraphs called Xrings. HypergraphHis an Xring if the hyperedges ofHcan be circularly ordered so that for every vertex, all hyperedges containing the vertex are consecutive; in addition, the vertices of no hyperedge may be a subset of the vertices of another hyperedge, and no vertex may appear in exactly one hyperedge. LetH₁andH₂be two hypergraphs. A tree projection ofH₁with respect toH₂is an acyclic hypergraphH₃such that the vertices of each hyperedge inH₁appear among the vertices of some hyperedge ofH₃and the vertices of each hyperedge inH₃appear among the vertices of some hyperedge ofH₂. A polynomial time algorithm is presented for deciding, given XringH₁and arbitrary hypergraphH₂, whether there exists a tree projection ofH₂with respect toH₁. It is shown that hypergraphHis an Xring iff a modified adjacency graph ofHis a circular-arc graph. A linear time Xring recognition algorithm, for GYO reduced hypergraphs as inputs, is presented.     

On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by x′ _Aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.     

Acyclic List Edge Coloring of Graphs

A proper edge coloring of a graph is said to be acyclic if any cycle is colored with at least three colors. An edge-list L of a graph G is a mapping that assigns a finite set of positive integers to each edge of G. An acyclic edge coloring ? of G such that for any is called an acyclic L-edge coloring of G. A graph G is said to be acyclically k-edge choosable if it has an acyclic L‐edge coloring for any edge‐list L that satisfies for each edge e. The acyclic list chromatic index is the least integer k such that G is acyclically k‐edge choosable. We develop techniques to obtain bounds for the acyclic list chromatic indices of outerplanar graphs, subcubic graphs, and subdivisions of Halin graphs.     

Acyclic edge colorings of graphs

A proper coloring of the edges of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

A sufficient condition for planar graphs to be acyclically 5‐choosable

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v)|v∈V} of G, we say G is acyclically L‐list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L‐list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k‐choosable. In this article we prove that every planar graph without 4‐cycles and without intersecting triangles is acyclically 5‐choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216–6225], which says that every planar graph without 4‐cycles and without two triangles at distance less than 3 is acyclically 5‐choosable. © 2011 Wiley Periodicals, Inc. J Graph Theory     

On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L‐list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper coloring c of G such that c (v) ∈ L(v) for all v ∈ V. If G is L‐list colorable for every list assignment with |L (v)| ≥ k for all v ∈ V, then G is said k‐choosable. A graph is said to be acyclically k‐choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k‐choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281–300, 2006     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

Acyclic edge chromatic number of outerplanar graphs

A proper edge coloring of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by χ(G), is the least number of colors in an acyclic edge coloring of G. In this paper, we determine completely the acyclic edge chromatic number of outerplanar graphs. The proof is constructive and supplies a polynomial time algorithm to acyclically color the edges of any outerplanar graph G using χ(G) colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 22–36, 2010     

Acyclic 5‐choosability of planar graphs without adjacent short cycles

The conjecture on acyclic 5‐choosability of planar graphs [Borodin et al., 2002] as yet has been verified only for several restricted classes of graphs. None of these classes allows 4‐cycles. We prove that a planar graph is acyclically 5‐choosable if it does not contain an i‐cycle adjacent to a j‐cycle where 3?j?5 if i = 3 and 4?j?6 if i = 4. This result absorbs most of the previous work in this direction. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:169‐176, 2011     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

Acyclick-strong coloring of maps on surfaces

A coloring of graph vertices is called acyclic if the ends of each edge are colored in distinct colors and there are no two-colored cycles. Suppose each face of rank not greater thank, k ≥ 4, on a surfaceS ^N is replaced by the clique on the same set of vertices. Then the pseudograph obtained in this way can be colored acyclically in a set of colors whose cardinality depends linearly onN and onk. Results of this kind were known before only for 1 ≤N ≤ 2 and 3 ≤k ≤ 4. Translated fromMatematicheskie Zametki, vol. 67, No. 1, pp. 36–45, January, 2000.     

Planar graphs without 4‐cycles are acyclically 6‐choosable

A proper vertex coloring of a graph G=(V, E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L‐list colorable if for a given list assignment L={L(v)|v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L‐list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k‐choosable. In this paper we prove that every planar graph G without 4‐cycles is acyclically 6‐choosable. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 307–323, 2009     

An upper bound for the adjacent vertex distinguishing acyclic edge chromatic number of a graph

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′ _aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′ _aa (G) ≤ 32Δ. Supported by the Natural Science Foundation of Gansu Province (3ZS051-A25-025)     

Acyclic 5‐choosability of planar graphs without small cycles

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L‐list colorable if for a given list assignment L = {L(v): v: ∈ V}, there exists a proper acyclic coloring ? of G such that ?(v) ∈ L(v) for all v ∈ V. If G is acyclically L‐list colorable for any list assignment with |L (v)|≥ k for all v ∈ V, then G is acyclically k‐choosable. In this article, we prove that every planar graph G without 4‐ and 5‐cycles, or without 4‐ and 6‐cycles is acyclically 5‐choosable. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 245–260, 2007     

The acyclic edge chromatic number of a random d‐regular graph is d + 1

We prove the theorem from the title: the acyclic edge chromatic number of a random d‐regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov, and Zaks and presents further support for a conjecture that Δ(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analog of a result of Robinson and the second author on edge chromatic number. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 69–74, 2005