Acyclic colorings of planar graphs

A coloring of the vertices of a graph byk colors is called acyclic provided that no circuit is bichromatic. We prove that every planar graph has an acyclic coloring with nine colors, and conjecture that five colors are sufficient. Other results on related types of colorings are also obtained; some of them generalize known facts about “point-arboricity”. Research supported in part by the Office of Naval Research under Grant N00014-67-A-0103-0003.     

Acyclic improper choosability of graphs

We consider improper colorings (sometimes called generalized, defective or relaxed colorings) in which every color class has a bounded degree. We propose a natural extension of improper colorings: acyclic improper choosability. We prove that subcubic graphs are acyclically (3, 1)*-choosable (i.e. they are acyclically 3-choosable with color classes of maximum degree one). Using a linear time algorithm, we also prove that outerplanar graphs are acyclically (2, 5)*-choosable (i.e. they are acyclically 2-choosable with color classes of maximum degree five). Both results are optimal. We finally prove that acyclic choosability and acyclic improper choosability of planar graphs are equivalent notions.     

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     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.     

Colorings and orientations of graphs

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).Research supported in part by a United States-Israel BSF Grant and by a Bergmann Memorial Grant.     

Acyclic edge colorings of planar graphs and seriesparallel 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 a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

Acyclic edge coloring of subcubic 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 it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

Acyclic colorings of locally planar graphs

It is proved that graphs embedded in a surface with large nonseparating edge-width can be acyclically 8-colored. The condition on large nonseparating edge-width is expressed in terms of non-null-homologous circuits and is a weaker requirement than asking for large edge-width (which is based on homotopy).     

Acyclic edge colouring of plane graphs

Let $G=(V,E)$ be any finite graph. A mapping $c:E\to \left[k\right]$ is called an acyclic edge $k$-colouring of $G$, if any two adjacent edges have different colours and there are no bichromatic cycles in $G$. The smallest number $k$ of colours, such that $G$ has an acyclic edge $k$-colouring is called the acyclic chromatic index of $G$, denoted by ${\chi }^{\prime }\left(G\right)$.In 1978, Fiam?ík conjectured that for any graph $G$ it holds ${\chi }^{\prime }\left(G\right)\le \Delta \left(G\right)+2$, where $\Delta \left(G\right)$ stands for the maximum degree of $G$. This conjecture has been verified by now only for some special classes of graphs. In 2010, Borowiecki and Fiedorowicz confirmed it for planar graph with girth at least 5. In this paper, we improve the above result, by proving that if $G$ is a plane graph such that for each pair $i,j\in \{3,4\}$, no $i$-face and a $j$-face share a common vertex in $G$, then ${\chi }^{\prime }\left(G\right)\le \Delta \left(G\right)+2$.     

Acyclic 4-choosability of planar graphs

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 paper we prove that planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.     

Acyclic directed graphs representing independence models

In this paper we study the problem of representing probabilistic independence models, in particular those closed under graphoid properties. We focus on acyclic directed graph (DAG): a new algorithm to build a DAG, given an ordering among random variables, is described and peculiarities and advantages of this approach are discussed. Moreover, we provide a necessary and sufficient condition for the existence of a perfect map representing an independence model and we describe an algorithm based on this characterization.     

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     

Completing orientations of partially oriented graphs

We initiate a general study of what we call orientation completion problems. For a fixed class of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in by orienting the (nonoriented) edges in P. Orientation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically representable graphs. We study orientation completion problems for various classes of oriented graphs, including k‐arc‐strong oriented graphs, k‐strong oriented graphs, quasi‐transitive‐oriented graphs, local tournaments, acyclic local tournaments, locally transitive tournaments, locally transitive local tournaments, in‐tournaments, and oriented graphs that have directed cycle factors. We show that the orientation completion problem for each of these classes is either polynomial time solvable or NP‐complete. We also show that some of the NP‐complete problems become polynomial time solvable when the input‐oriented graphs satisfy certain extra conditions. Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable. The latter generalizes a previous result.     

Counting restricted orientations of random graphs

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.     

Minimal orientations of colour critical graphs

In 1966 T. Gallai asked whether every criticalk-chromatic graph possesses an orientation having just one directed path of lengthk–1. In this note we show that in general the answer is negative, but also that the answer is affirmative whenk5 and the graph has maximal degree at mostk.