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 Fiam{č}ik (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.
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…
A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $χ'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. The maximum average degree of a graph $G$, denoted by $\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In this paper, it is proved that if $\mad(G)<4$, then $χ'_a(G)\leq{Δ(G)+2}$; if $\mad(G)<3$, then $χ'_a(G)\leq{Δ(G)+1}$. This implies that every triangle-free planar graph $G$ is acyclically edge $(Δ(G)+2)$-colorable.
An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $Δ= Δ(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = Δ+ 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus es…
An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star ch…
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 $\chiup'_{a}(G)\leq Δ(G) + 2$ 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.
Proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $χ'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. Basavaraju et al. [Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2) (2011), 463--478] showed that $χ'_a(G)\le Δ(G)+12$ for planar graphs $G$ with maximum degree $Δ(G)$. In this paper, the bound is improved to $Δ(G)+10$.
An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiamčík (1978) conjectured that $\chiup_{a}'(G) \leq Δ(G) + 2$, where $Δ(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $κ$ is {\em $κ$-deletion-minimal} if $\ch…
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 $\chiup'_{a}(G)\leq Δ(G) + 2$ for any simple graph $G$ with maximum degree $Δ(G)$. 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 $Δ(G) + 16$ color…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{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 $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. We prove that if $G$ is a planar graph with maximum degree $Δ$, then $a'(G)\le Δ+ 12$.
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. Fiam${\rm \check{c}}$ik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that $a'(G)\le Δ+ 2$ for any simple graph $G$ with maximum degree $Δ$. Basavaraju and Chandran (2009) showed that every graph $G$ with $Δ=4$, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular g…
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 $\lceil 3.74(Δ-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(Δ-1)$. The main contribution of this work is that it comprises a probabilistic an…
A graph is called 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, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Therefore, two open problems presented by Fabrici and Madaras [The structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are solved. Furthermore, we prove that each 1-planar graph $G$ with …
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{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 $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. If every induced subgraph $H$ of $G$ satisfies the condition $\vert E(H) \vert \le 2\vert V(H) \vert -1$, we say that the graph $G$ satisfies $Property\ A$. In this paper, we prove that…
An edge labeling of a graph distinguishes neighbors by sets (multisets, resp.), if for any two adjacent vertices $u$ and $v$ the sets (multisets, resp.) of labels appearing on edges incident to $u$ and $v$ are different. In an analogous way we define total labelings distinguishing neighbors by sets or multisets: for each vertex, we consider labels on incident edges and the label of the vertex itself. In this paper we show that these problems, and also other problems of similar flavor, admit an elegant and natural generalization as a hypergraph coloring problem. An ieds-coloring (iedm-coloring,…
Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, the works of Grytczuk et al. on words, and Dujmović et al. on colorings, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called \emph{entropy compression method}. Inspired by this work, we propose a more general framework and a better analysis. This leads to improved upper bounds on chromatic numbers and indices. In particular, every graph with maximum degree $Δ$ has an a…
Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, inc…
In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals is allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains $k$-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of $O(…
In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number, the Thue chromatic number, the harmonious chromatic number and the clique chromatic number of $P_4$-tidy graphs and $(q,q-4)$-graphs, for every fixed $q$. These classes include cographs, $P_4$-sparse and $P_4$-lite graphs. All these coloring problems are known to be NP-hard for general graphs. These algorithms are fixed parameter tractable on the parameter $q(G)$, which is the minimum $q$ such that $G$ is a $(q,q-4)$-graph. We also prove that every connected $(q,q-4)$-graph…
In the present paper, we study algorithmic questions for the arc-intersection graph of directed paths on a tree. Such graphs are known to be perfect (proved by Monma and Wei in 1986). We present faster algorithms than all previously known algorithms for solving the minimum coloring and the minimum clique cover problems. They both run in $O(np)$ time, where $n$ is the number of vertices of the tree and $p$ the number of paths. Another result is a polynomial algorithm computing a kernel in the intersection graph, when its edges are oriented in a clique-acyclic way. Indeed, such a kernel exists f…
We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $η_t…
Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdur…
Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most $3$. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which ma…
In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS). We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on res…
