Acyclic Edge Research Articles

Acyclic chromatic index of chordless graphs

An acyclic edge coloring of a graph is a proper edge coloring with no bichromatic cycles. The acyclic chromatic index of a graph G denoted by a′(G), is the minimum integer k such that G has an acyclic edge coloring with k colors. It was conjectured by Fiamčík [13] that a′(G)≤Δ+2 for any graph G with maximum degree Δ. Linear arboricity of a graph G, denoted by la(G), is the minimum number of linear forests into which the edges of G can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. By a result of Basavaraju and Chandran [6], if G is chordless, then a′(G)≤Δ+1. Machado, de Figueiredo and Trotignon [23] proved that the chromatic index of a chordless graph is Δ when Δ≥3. We prove that for any chordless graph G, a′(G)=Δ, when Δ≥3. Notice that this is an improvement over the result of Machado et al., since any acyclic edge coloring is also a proper edge coloring and we are using the same number of colors. As a byproduct, we prove that la(G)=⌈Δ2⌉, when Δ≥3. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado et al. [23] in case of chromatic index. This might be of independent interest.

Acyclic edge coloring of planar graphs

<abstract><p>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 $.</p></abstract>

Acyclic Edge Coloring of Chordal Graphs with Bounded Degree

Acyclic Edge Coloring of Chordal Graphs with Bounded Degree

Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiamčik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree Δ is acyclically edge (Δ+2)-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.

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.

Acyclic Edge Coloring of IC-planar Graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G is the least number of colors such that G has an acyclic edge coloring and denoted by Χ′a(G). An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. In this paper, it is proved that Χ′a(G) ≤ Δ(G) + 10, if G is an IC-planar graph without adjacent triangles and Χ′a(G) ≤ Δ(G) + 8, if G is a triangle-free IC-planar graph.

A new bound on the acyclic edge chromatic number

In this note we obtain a new bound for the acyclic edge chromatic number a′(G) of a graph G with maximum degree Δ proving that a′(G)≤3.569(Δ−1). To get this result we revisit and slightly modify the method described in Kirousis et al. (2017).

Approximation algorithms for capacitated partial inverse maximum spanning tree problem

Given an edge weighted graph, and an acyclic edge set, the goal of the partial inverse maximum spanning tree problem is to modify the weight function as little as possible such that there exists a maximum spanning tree with respect to the new weight function containing the given edge set. In this paper, we consider this problem with capacitated constraint under the $$l_{p}$$-norm, where p is an integer and $$p \in [1,+\,\infty )$$. Firstly, we characterize the feasible solutions of this problem. Then, we present a $$\root p \of {k}$$-approximation algorithm for this problem when the weight function can only be decreased, where k is the number of edges in the given edge set. Finally, when the weight function can be either decreased and increased, we propose an approximation algorithm for the general case and analyse its approximation ratio. Moreover, we remark that these algorithms can be generalized under the weighted $$l_{p}$$-norm and the weighted sum Hamming distance.

Capacitated partial inverse maximum spanning tree under the weighted Hamming distance

Given an edge weighted graph, and an acyclic edge set, the goal of partial inverse maximum spanning tree problem is to modify the weight function as little as possible such that there exists a maximum spanning tree with respect to the new weight function containing the given edge set. In this paper, we consider this problem with capacitated constraint under the weighted Hamming distance. Under the weighted sum Hamming distance, if the given edge set has at least two edges, we show that this problem is APX-Hard even without the capacitated constraint; if the given edge set contains only one edge, we present a strongly polynomial time algorithm to solve it. Under the weighted bottleneck Hamming distance, we present an algorithm with time complexity \(O(m\log ^2 m)\), where m is the number of edges of the given graph.

Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$ l p -norm

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.

Constant-Time Local Computation Algorithms

Local computation algorithms (LCAs) produce small parts of a single (possibly approximate) solution to a given search problem using time and space sublinear in the size of the input. In this work we present LCAs whose time complexity (and usually also space complexity) is independent of the input size. Specifically, we give (1) a (1 − 𝜖)-approximation LCA to the maximum weight acyclic edge set, (2) LCAs for approximating multicut and integer multicommodity flow on trees, and (3) a local reduction of weighted matching to any unweighted matching LCA, such that the running time of the weighted matching LCA is d times (where d is the maximal degree) the running time of the unweighted matching LCA, (and therefore independent of the edge weight function).

Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs

‎Let $G$ be a graph and $chi^{prime}_{aa}(G)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $G$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎We prove a general bound for $chi^{prime}_{aa}(Gsquare H)$ for any two graphs $G$ and $H$‎. ‎We also determine‎ ‎exact value of this parameter for the Cartesian product of two paths‎, ‎Cartesian product of a path and a cycle‎, ‎Cartesian product of two trees‎, ‎hypercubes‎. ‎We show that $chi^{prime}_{aa}(C_msquare C_n)$ is at most $6$ fo every $mgeq 3$ and $ngeq 3$‎. ‎Moreover in some cases we find the exact value of $chi^{prime}_{aa}(C_msquare C_n)$‎.

Acyclic Chromatic Index of Triangle-free 1-Planar Graphs

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq \Delta(G) + 2$ for any simple graph $G$ with maximum degree $\Delta(G)$. A graph is {\em $1$-planar} if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we prove that every triangle-free $1$-planar graph $G$ has an acyclic edge coloring with $\Delta(G) + 16$ colors.

Acyclic edge coloring through the Lovász Local Lemma

We give a probabilistic analysis of a Moser-type algorithm for the Lovász Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree Δ has an acyclic proper edge coloring with at most ⌈3.74(Δ−1)⌉+1 colors, whereas, previously, the best bound was 4(Δ−1). The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.

Acyclic edge colourings of graphs with large girth

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)\Delta$ colours.

New bounds for the acyclic chromatic index

An edge coloring of a graph G is called an acyclic edge coloring if it is proper and every cycle in G contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of G is called the acyclic chromatic index of G and is denoted by a′(G). Fiamčik (1978) and independently Alon, Sudakov, and Zaks (2001) conjectured that a′(G)≤Δ(G)+2, where Δ(G) denotes the maximum degree of G. The best known general bound is a′(G)≤4(Δ(G)−1) due to Esperet and Parreau (2013). We apply a generalization of the Lovász Local Lemma to show that if G contains no copy of a given bipartite graph H, then a′(G)≤3Δ(G)+o(Δ(G)). Moreover, for every ε>0, there exists a constant c such that if g(G)≥c, then a′(G)≤(2+ε)Δ(G)+o(Δ(G)), where g(G) denotes the girth of G.

Acyclic edge coloring of triangle-free 1-planar graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ′ a (G), is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that χ′ a (G) ≤ Δ(G)+22, if G is a triangle-free 1-planar graph.

Further result on acyclic chromatic index of planar graphs

An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index χa′(G) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that χa′(G)≤Δ(G)+2 for any simple graph G with maximum degree Δ(G). In this paper, we prove that every planar graph G admits an acyclic edge coloring with Δ(G)+6 colors.

A New Approach to Compute Acyclic Chromatic Index of Certain Chemical Structures

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 G denoted by ) ( ' G a  is the minimum number k such that there is an acyclic edge coloring using k colors. The maximum degree in G denoted by ∆(G), is the lower bound for ) ( ' G a  . Pcuts introduced in this paper

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.