Acyclic Coloring Research Articles

Defective acyclic colorings of planar graphs

AbstractThis paper studies two variants of defective acyclic coloring of planar graphs. For a graph and a coloring of , a 2‐colored cycle (2CC) transversal is a subset of that intersects every 2‐colored cycle. Let be a positive integer. We denote by the minimum integer such that has a proper ‐coloring which has a 2CC transversal of size , and by the minimum size of a subset of such that is acyclic ‐colorable. We prove that for any ‐vertex 3‐colorable planar graph and for any planar graph provided that . We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal can be chosen in such a way that induces a forest. We also prove that for any planar graph and .

A Note on Acyclic Coloring of Strong Product of Graphs

A Note on Acyclic Coloring of Strong Product of Graphs

Hardness transitions and uniqueness of acyclic colouring

For k∈N, a k-acyclic colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that (i) f(u)≠f(v) for every edge uv of G, and (ii) there is no cycle in G bicoloured by f. For k∈N, the problem k-Acyclic Colourability takes a graph G as input and asks whether G admits a k-acyclic colouring. Ochem (EuroComb 2005) proved that 3-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree 4. Mondal et al. (2013) proved that 4-Acyclic Colourability is NP-complete for graphs of maximum degree five. We prove that for k≥3, k-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree k+1, thereby generalising the NP-completeness result of Ochem, and adding bipartiteness to the NP-completeness result of Mondal et al.. In contrast, k-Acyclic Colourability is polynomial-time solvable for graphs of maximum degree at most 0.38k3/4. Hence, for k≥3, the least integer d such that k-Acyclic Colourability in graphs of maximum degree d is NP-complete, denoted by La(k), satisfies 0.38k3/4<La(k)≤k+1. We prove that for k≥4, k-Acyclic Colourability in d-regular graphs is NP-complete if and only if La(k)≤d≤2k−3. We also show that it is coNP-hard to check whether an input graph G admits a unique k-acyclic colouring up to colour swaps (resp. up to colour swaps and automorphisms).

Application of acyclic coloring in wavelength assignment problem for butterfly and Beneš network

Application of acyclic coloring in wavelength assignment problem for butterfly and Beneš network

Acyclic Coloring of Certain Graphs

A graph G with acyclic coloring has no two adjacent vertices with the same color and no bichromatic cycle. Also, the coloring results in a forest when any two-color classes are combined. The concept of acyclic coloring plays a pivotal role in the computation of Hessians, Kekule structures classification, coding theory, and statistical mechanics. In this paper, the acyclic chromatic number of generalized fan graph, generalized Möbius ladder graph and flower snark graph have been determined.

On b-acyclic chromatic number of a graph

Let G be a graph. We introduce the acyclic b-chromatic number of G as an analogue to the b-chromatic number of G. An acyclic coloring of a graph G is a map c:V(G)rightarrow {1,ldots ,k} such that c(u)ne c(v) for any uvin E(G) and the induced subgraph on vertices of any two colors i,jin {1,ldots ,k} induces a forest. On the set of all acyclic colorings of G we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number A(G) of G and the maximum cardinality of its minimal element is the acyclic b-chromatic number A_{textrm{b}}(G) of G. We present several properties of A_{textrm{b}}(G). In particular, we derive A_{textrm{b}}(G) for several known graph families, derive some bounds for A_{textrm{b}}(G), compare A_{textrm{b}}(G) with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.

Acyclic coloring of claw-free graphs with small degree

An acyclic k -coloring of a graph is a proper vertex coloring using k colors such that every cycle is colored with at least three colors. A graph is claw-free if it contains no K 1 , 3 as an induced subgraph. Grünbaum (1973) conjectured that every graph with maximum degree Δ has an acyclic ( Δ + 1 ) -coloring. In this paper, we show that this conjecture holds for the claw-free graphs with Δ ≤ 6 .

Acyclic and Star Coloring of Powers of Paths and Cycles

Let $G=(V,E)$ be a graph. The $k^{th}-$ power of $G$ denoted by $G^{k}$ is the graph whose vertex set is $V$ and in which two vertices are adjacent if and only if their distance in $G$ is at most $k.$ A vertex coloring of $G$ is acyclic if each bichromatic subgraph is a forest. A star coloring of $G$ is an acyclic coloring in which each bichromatic subgraph is a star forest.The minimum number of colors such that $G$ admits an acyclic (star) coloring is called the acyclic (star) chromatic number of G, and denoted by $\chi_{a}(G)(\chi_{s}(G))$. In this paper we prove that for $n\geq k+1,$ $\chi_{s}(P_{n}^{k})=\min\{\lfloor\frac{k+n+1}{2}\rfloor,2k+1\}$ and $\chi_{a}(P_{n}^{k})=k+1.$ Further we show that for $n\geq(k+1)^{2}$, $%2k+1\leq\chi _{s}(C_{n}^{k})\leq2k+2$ and $k+2\leq\chi_{a}(C_{n}^{k})\leq k+3.$ Finally, we derive the formula $\chi_{a}(C_{n}^{k})=k+2$ for $% n\geq(k+1)^{3}.$

Acyclic, Star, and Injective Colouring: Bounding the Diameter

We examine the effect of bounding the diameter for a number of natural and well-studied variants of the COLOURING problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are ACYCLIC COLOURING, STAR COLOURING and INJECTIVE COLOURING. The last problem is also known as $L(1,1)$-LABELLING and we also consider the framework of $L(a,b)$-LABELLING. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most~$d$, ACYCLIC $3$-COLOURING is polynomial-time solvable if $d\leq 2$ but NP-complete if $d\geq 4$, and STAR $3$-COLOURING is polynomial-time solvable if $d\leq 3$ but NP-complete for $d\geq 8.$ As far as we are aware, STAR $3$-COLOURING is the first problem that exhibits a complexity jump for some $d\geq 3.$ Our third main result is that $L(1,2)$-LABELLING is NP-complete for graphs of diameter~$2$; we relate the latter problem to a special case of HAMILTONIAN PATH.

Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

AbstractWe show that for any fixed integer , a graph of maximum defiggree has a coloring with colors in which every connected bicolored subgraph contains at most edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which colors suffice, and acyclic colorings, for which colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree has a proper coloring with colors in which every bicolored subgraph is planar, as well as a proper coloring with colors in which every bicolored subgraph has treewidth at most 3.

On star and acyclic coloring of generalized lexicographic product of graphs

&lt;abstract&gt;&lt;p&gt;A $ star \; coloring $ of a graph $ G $ is a proper vertex coloring of $ G $ such that any path of length 3 in $ G $ is not bicolored. The $ star \; chromatic \; number $ $ \chi_s(G) $ of $ G $ is the smallest integer $ k $ for which $ G $ admits a star coloring with $ k $ colors. A $ acyclic \; coloring $ of $ G $ is a proper coloring of $ G $ such that any cycle in $ G $ is not bicolored. The $ acyclic \; chromatic \; number $ of $ G $, denoted by $ a(G) $, is the minimum number of colors needed to acyclically color $ G $. In this paper, we present upper bound for the star and acyclic chromatic numbers of the generalized lexicographic product $ G[h_n] $ of graph $ G $ and disjoint graph sequence $ h_n $, where $ G $ exists a $ k- $colorful neighbor star coloring or $ k- $colorful neighbor acyclic coloring. In addition, the upper bounds are tight.&lt;/p&gt;&lt;/abstract&gt;

Acyclic edge coloring of planar graphs

&lt;abstract&gt;&lt;p&gt;An acyclic edge coloring of a graph $ G $ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index of $ G $, denoted by $ \chi^{'}_{a}(G) $, is the smallest integer $ k $ such that $ G $ is acyclically edge $ k $-colorable. In this paper, we consider the planar graphs without 3-cycles and intersecting 4-cycles, and prove that $ \chi^{'}_{a}(G)\leq\Delta(G)+1 $ if $ \Delta(G)\geq 8 $.&lt;/p&gt;&lt;/abstract&gt;

Acyclic Edge Coloring of Chordal Graphs with Bounded Degree

Acyclic Edge Coloring of Chordal Graphs with Bounded Degree

Local conditions for planar graphs of acyclic edge coloring

Give a graph G, we color its all edges. If any two adjacent edges gets the different colors, then we call this color a proper edge coloring of G. Give a proper edge coloring of G, if only two colors alternately appear on a cycle, then the cycle is called bichromatic. Acyclic edge coloring of a graph G means that there are no bichromatic cycles in G. The acyclic chromatic index of a graph G is the minimum number k such that G has an acyclic edge coloring using k colors. Denoted $${\chi ^{'}_a}(G)$$ as the acyclic chromatic index of G. A planar graph is a graph that can be embedded in the plane in such a way that no two edges intersect geometrically except at a vertex to which they are both incident. In this paper, we use the discharging method to prove that $${\chi ^{'}_a}(G)\le \varDelta (G)+ 2$$ if G is a planar graph and there is an integer $$k_v \in \{3, 4, 5\}$$ such that v is not contained in any $$k_v$$ -cycle for every vertex v.

On Star Coloring of Several Corona Graphs

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs (G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi |i = 1,2,⋯m}∪{vij |i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi ). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs (G ∘ H) and the exact values of χs (G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs (G ∘ H) ≤ χs (G) + χs (H); χs (Pm ∘ H) = χs (H) + 2; χs (K 1,m ∘ H) = χs (H) + 2; χs (Cn ∘ H) = χs (H) + 2, where n ≠ 5.

Acyclic Coloring of Graphs with Maximum Degree 7

The acyclic chromatic number a(G) of a graph G is the minimum number of colors such that G has a proper vertex coloring and no bichromatic cycles. For a graph G with maximum degree $$\Delta $$ , Grunbaum (1973) conjectured $$a(G)\le \Delta +1$$ . Up to now, the conjecture has only been shown for $$\Delta \le 4$$ . In this paper, it is proved that $$a(G)\le 12$$ for $$\Delta =7$$ , thus improving the result $$a(G)\le 17$$ of Dieng et al. (in: Proc. European conference on combinatorics, graph theory and applications, 2010).

Facets based on cycles and cliques for the acyclic coloring polytope

A coloring of a graph is an assignment of colors to its vertices such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring is a coloring such that no cycle receives exactly two colors, and the acyclic chromatic number χA(G) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determining whether χA(G) ≤ k or not is NP-complete even for k = 3. The acyclic coloring problem arises in the context of efficient computations of sparse and symmetric Hessian matrices via substitution methods. In a previous work we presented facet-inducing families of valid inequalities based on induced even cycles for the polytope associated to an integer programming formulation of the acyclic coloring problem. In this work we continue with this study by introducing new families of facet-inducing inequalities based on combinations of even cycles and cliques.

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.

Acyclic coloring of graphs and entropy compression method

Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, Dujmović et al. (2016) initiated the use of the so-called entropy compression method for graph coloring problems. Then, using the same approach Esperet and Parreau (2013) proved new upper bounds for several chromatic numbers, and explained how that approach could be used for many different coloring problems. Here, we follow this line of research for the particular case of acyclic coloring: we show that every graph with maximum degree Δ has acyclic chromatic number at most 32Δ43+O(Δ).

Acyclic coloring of IC-planar graphs

A graph G is called an IC-planar graph if it can be embedded in the plane so that every edge is crossed by at most one other edge and every vertex is incident to at most one crossing edge. In this paper, we prove that every IC-planar graph is acyclically 10-colorable. Moreover, an IC-planar graph of the acyclic chromatic number 6 is constructed.