Acyclic Coloring Research Articles

Acyclic [formula omitted]-coloring of graphs with maximum degrees 5 and 6

For a graph G=(V(G),E(G)), an acyclic coloring of G is a proper vertex coloring such that every cycle is colored with at least three colors. A graph G is acyclically L-colorable if for a given list assignment L={L(v)|v∈V(G)}, there exists an acyclic coloring f such that f(v)∈L(v) for all v∈V(G). If G is acyclically L-colorable for every list assignment L with |L(v)|≥k for all v∈V(G), then G is acyclically k-choosable. In this paper, we show that every graph with maximum degree 5 is acyclically 7-choosable and every graph with maximum degree 6 is acyclically 10-choosable.

Acyclic coloring of graphs with maximum degree at most six

An acyclic k-coloring of a graph G is a proper vertex coloring using k colors such that any cycle of G is colored with at least three colors, and the minimum integer k is called the acyclic chromatic number of G, denoted by χa(G). Grünbaum conjectured that the acyclic chromatic number of graphs with maximum degree at most Δ is no more than Δ+1 in 1973. By now, the conjecture has been proved only for Δ≤4. We show that χa(G)≤9 for Δ≤6, which improves the result χa(G)≤10 of Zhao et al. (2014).

Characterization of forbidden subgraphs for bounded star chromatic number

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are 2-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an H-avoidingk-coloring to be a proper k-coloring that forbids a bicolored subgraph H.When considering the class C of graphs with no F as an induced subgraph, it is not hard to see that every graph in C has bounded chromatic number if and only if F is a complete graph of size at most two. We study this phenomena for the class of graphs with no F as a subgraph for H-avoiding coloring. We completely characterize all graphs F where the class of graphs with no F as a subgraph has bounded H-avoiding chromatic number for a large class of graphs H. As a corollary, our main result implies a characterization of graphs F where the class of graphs with no F as a subgraph has bounded star chromatic number. We also obtain a complete characterization for the acyclic chromatic number.

The Diachromatic Number of Digraphs

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromaticnumber is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.

Construction of acyclically 4-colourable planar triangulations with minimum degree 4

ABSTRACTAn acyclic colouring of a graph is a proper colouring of the graph, for which every cycle uses at least three colours. In this paper, we describe a method for constructing all 4-connected acyclically 4-colourable planar triangulations that have exactly four odd-vertices, except the ones that contain no adjacent odd-vertices. Unlike previous operations, our method successfully establishes a connection with (acyclic) 4-colourings. Moreover, we discuss a special class of graphs, called diamond triangulations, and give a necessary and sufficient condition for a diamond triangulation to be acyclically 4-colourable.

On acyclically 4-colorable maximal planar graphs

An acyclic coloring of a graph is a proper coloring of the graph, for which every cycle uses at least three colors. Let G4 be the set of maximal planar graphs of minimum degree 4, such that each graph in G4 contains exactly four odd-vertices and the subgraph induced by the four odd-vertices contains a quadrilateral. In this article, we show that every acyclic 4-coloring of a maximal planar graph with exact four odd-vertices is locally equitable with regard to its four odd-vertices. Moreover, we obtain a necessary and sufficient condition for a graph in G4 to be acyclically 4-colorable, and give an enumeration of the acyclically 4-colorable graphs in G4.

On Acyclic Coloring of Mycielskians

An acyclic coloring of a graph \(G\) is a proper vertex coloring such that the induced subgraph of any two color classes is acyclic. The minimum number of colors needed to acyclically color the vertices of a graph \(G\) is called as acyclic chromatic number and is denoted by \(\chi_a(G)\). In this paper, we give the exact value of the acyclic chromatic number of Mycielskian graph of cycles, paths, complete graphs and complete bipartite graphs.

Plurigraph Coloring and Scheduling Problems

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contraction law can be applied to scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exists pairs of 3-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.

Universal targets for homomorphisms of edge-colored graphs

A k-edge-colored graph is a finite, simple graph with edges labeled by numbers 1,…,k. A function from the vertex set of one k-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class F of graphs, a k-edge-colored graph H (not necessarily with the underlying graph in F) is k-universal for F when any k-edge-colored graph with the underlying graph in F admits a homomorphism to H. We characterize graph classes that admit k-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph.For a nonempty graph G, the density of G is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of G. For a nonempty class F of graphs, D(F) denotes the density of F, that is the supremum of densities of graphs in F.The main results are the following. The class F admits k-universal graphs for k⩾2 if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in F. For any such class, there exists a constant c, such that for any k⩾2, the size of the smallest k-universal graph is between kD(F) and ck⌈D(F)⌉.A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (1998) [2]. One of their results is that for the class of planar graphs, the size of the smallest k-universal graph is between k3+3 and 5k4. Our results yield that there exists a constant c such that for all k, this size is bounded from above by ck3.

Colourings, homomorphisms, and partitions of transitive digraphs

We investigate the complexity of generalizations of colourings (acyclic colourings, (k,ℓ)-colourings, homomorphisms, and matrix partitions), for inputs restricted to the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.

Strongly polynomial sequences as interpretations

A strongly polynomial sequence of graphs (Gn) is a sequence (Gn)n∈N of finite graphs such that, for every graph F, the number of homomorphisms from F to Gn is a fixed polynomial function of n (depending on F). For example, (Kn) is strongly polynomial since the number of homomorphisms from F to Kn is the chromatic polynomial of F evaluated at n. In earlier work of de la Harpe and Jaeger, and more recently of Averbouch, Garijo, Godlin, Goodall, Makowsky, Nešetřil, Tittmann, Zilber and others, various examples of strongly polynomial sequences and constructions for families of such sequences have been found, leading to analogues of the chromatic polynomial for fractional colourings and acyclic colourings, to choose two interesting examples.We give a new model-theoretic method of constructing strongly polynomial sequences of graphs that uses interpretation schemes of graphs in more general relational structures. This surprisingly easy yet general method encompasses all previous constructions and produces many more. We conjecture that, under mild assumptions, all strongly polynomial sequences of graphs can be produced by the general method of quantifier-free interpretation of graphs in certain basic relational structures (essentially disjoint unions of transitive tournaments with added unary relations). We verify this conjecture for strongly polynomial sequences of graphs with uniformly bounded degree.

Acyclic colorings of graphs with bounded degree

A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it 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. In the paper we consider some generalised acyclic $k$-colourings, namely, we require that each colour class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic $5$-colouring such that each colour class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph $G$ has an acyclic 2-colouring in which each colour class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic $t$-improper colouring of any graph with maximum degree $d$ assuming that the number of colors is large enough.

Acyclic and star coloring of P4-reducible and P4-sparse graphs

An acyclic coloring of a graph G is a proper vertex coloring such that G contains no bicolored cycles. The more restricted notion of star coloring of G is an acyclic coloring in which each path of length 3 is not bicolored. In this paper, we mainly study on the acyclic and star coloring of P4-reducible and P4-sparse graphs. Moreover, we list polynomial-time algorithms for giving an optimal acyclic or star coloring of a P4-reducible or P4-sparse graph.

Acyclic coloring of graphs without bichromatic long path

Let G be a graph of maximum degree Δ. A proper vertex coloring of G is acyclic if there is no bichromatic cycle. It was proved by Alon et al. [Acyclic coloring of graphs. Random Structures Algorithms, 1991, 2(3): 277–288] that G admits an acyclic coloring with O(Δ4/3) colors and a proper coloring with O(Δ(k-1)/(k-2)) colors such that no path with k vertices is bichromatic for a fixed integer k ≥ 5. In this paper, we combine above two colorings and show that if k ≥ 5 and G does not contain cycles of length 4, then G admits an acyclic coloring with O(Δ(k-1)/(k-2)) colors such that no path with k vertices is bichromatic.

Acyclic coloring of graphs with some girth restriction

A vertex coloring of a graph $$G$$G is called acyclic if it is a proper vertex coloring such that every cycle $$C$$C receives at least three colors. The acyclic chromatic number of $$G$$G is the least number of colors in an acyclic coloring of $$G$$G. We prove that acyclic chromatic number of any graph $$G$$G with maximum degree $$\Delta \ge 4$$Δ?4 and with girth at least $$4\Delta $$4Δ is at most $$12\Delta $$12Δ.

A Note on Acyclic Edge Colouring of Star Graph Families

A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.

On Coloring of Fish Tail Graph

We present a coloring, star coloring, acyclic coloring of fish tail graph in this paper and also given relation between the above mention graph. We illustrate the 3 types of coloring by unique way.

On Coloring of Fish Tail Graph

We present a coloring, star coloring, acyclic coloring of fish tail graph in this paper and also given relation between the above mention graph. We illustrate the 3 types of coloring by unique way.

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 degree 4 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.

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors. The acyclic chromatic index of a graph G, denoted by a′(G), is the minimum number k such that there is an acyclic edge coloring using k colors. It is known that a′(G) ≤ 16Δ for every graph G where Δ denotes the maximum degree of G. We prove that a′(G) < 13.8Δ for an arbitrary graph G. We also reduce the upper bounds of a′(G) to 9.8Δ and 9Δ with girth 5 and 7, respectively.