The circular chromatic number of a digraph

We introduce the circular chromatic number χ_c of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k‐cycle has circular chromatic number k/(k – 1), for k ≥ 2, values of χ_c between 1 and 2 are possible. We show that in fact, χ_c takes on all rational values greater than 1. Furthermore, there exist digraphs of arbitrarily large digirth and circular chromatic number. It is NP‐complete to decide if a given digraph has χ_c at most 2. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 227–240, 2004     

Snarks with given real flow numbers

We show that for each rational number r such that 4<r?5 there exist infinitely many cyclically 4‐edge‐connected cubic graphs of chromatic index 4 and girth at least 5—that is, snarks—whose flow number equals r. This answers a question posed by Pan and Zhu [Construction of graphs with given circular flow numbers, J Graph Theory 43 [2003], 304–318]. © 2011 Wiley Periodicals, Inc. J Graph Theory 68: 189‐201, 2011     

The circular chromatic number of series‐parallel graphs

In this article, we consider the circular chromatic number χ_c(G) of series‐parallel graphs G. It is well known that series‐parallel graphs have chromatic number at most 3. Hence, their circular chromatic numbers are at most 3. If a series‐parallel graph G contains a triangle, then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a series‐parallel graph G has girth at least 2 ⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). The special case k = 2 of this result implies that a triangle free series‐parallel graph G has circular chromatic number at most 8/3. Therefore, the circular chromatic number of a series‐parallel graph (and of a K₄‐minor free graph) is either 3 or at most 8/3. This is in sharp contrast to recent results of Moser [5] and Zhu [14], which imply that the circular chromatic number of K₅‐minor free graphs are precisely all rational numbers in the interval [2, 4]. We shall also construct examples to demonstrate the sharpness of the bound given in this article. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 14–24, 2000     

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K_{⌈t/2⌉,⌊t/2⌋} in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme. * Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826, and NK62321. † Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046234, AT048826, and NK62321.     

Orientations of graphs with uncountable chromatic number

Motivated by an old conjecture of P. Erdős and V. Neumann‐Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that, consistently, there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected case: any graph with uncountable chromatic number contains a 4‐cycle. Next, we prove that several well‐known graphs (uncountable complete graphs, certain comparability graphs, and shift graphs) admit orientations with uncountable dichromatic number in ZFC. However, we show that the statement “every graph G of size and chromatic number ω₁ has an orientation D with uncountable dichromatic number” is independent of ZFC. We end the article with several open problems.     

The star-chromatic number of planar graphs

The star-chromatic number of a graph, a parameter introduced by Vince, is a natural generalization of the chromatic number of a graph. Here we construct planar graphs with star-chromatic number r, where r is any rational number between 2 and 3, partially answering a question of Vince. © 1997 John Wiley & Sons, Inc.     

Chromaticity of triangulated graphs

It is shown here that a connected graph G without subgraphs isomorphic to K₄ is triangulated if and only if its chromatic polynomial P(G,λ) equals λ(λ ? 1)^m(λ ? 2)^r for some integers m ≧ 1, r ≧ 0. This result generalizes the characterization of Two-Trees given by E.G. Whitehead [“Chromaticity of Two-Trees,” Journal of Graph Theory 9 (1985) 279–284].     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010     

Lightness of digraphs in surfaces and directed game chromatic number

The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum in-degree at least 1 (resp., graphs with minimum degree at least 2) and a given girth k, and without 4-cycles, which can be embedded in a surface S. (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k≥5, these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307-317] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game colouring number, and thus for the game chromatic number of (di)graphs with girth k and without 4-cycles embeddable in S. The game chromatic resp. game colouring number were introduced by Bodlaender [H.L. Bodlaender, On the complexity of some coloring games, Int. J. Found. Comput. Sci. 2 (1991) 133-147] resp. Zhu [X. Zhu, The game coloring number of planar graphs, J. Combin. Theory B 75 (1999) 245-258] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game colouring number of a directed simple forest is at most 3.     

Circular choosability is rational

The circular choosability or circular list chromatic number of a graph is a list-version of the circular chromatic number, that was introduced by Mohar in 2002 and has been studied by several groups of authors since then. One of the nice properties that the circular chromatic number enjoys is that it is a rational number for all finite graphs G, and a fundamental question, posed by Zhu and reiterated by others, is whether the same holds for the circular choosability. In this paper we show that this is indeed the case.     

A note on the girth of digraphs

Behzad, Chartrand and Wall conjectured that the girth of a diregular graph of ordern and outdegreer is not greater than [n /r]. This conjecture has been proved forr=2 by Behzad and forr=3 by Bermond. We prove that a digraph of ordern and halfdegree ≧4 has girth not exceeding [n / 4]. We also obtain short proofs of the above results. Our method is an application of the theory of connectivity of digraphs.     

Circular Coloring of Planar Digraphs

Let D be a digraph. The circular chromatic number ${\chi_c(D)}$ and chromatic number ${\chi(D)}$ of D were proposed recently by Bokal et?al. Let ${\vec{\chi_c}(G)={\rm max}\{\chi_c(D)| D\, {\rm is\, an\, orientation\, of} G\}}$ . Let G be a planar graph and n?≥ 2. We prove that if the girth of G is at least ${\frac{10n-5}{3},}$ then ${\vec{\chi_c}(G)\leq \frac{n}{n-1}}$ . We also study the circular chromatic number of some special planar digraphs.     

Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs

A graph is planar if it can be embedded on the plane without edge-crossings. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239–249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215–219, 2007].     

On the Hardness of Approximating the Chromatic Number

k -colorable for some fixed . Our main result is that it is NP-hard to find a 4-coloring of a 3-chromatic graph. As an immediate corollary we obtain that it is NP-hard to color a k-chromatic graph with at most colors. We also give simple proofs of two results of Lund and Yannakakis [20]. The first result shows that it is NP-hard to approximate the chromatic number to within for some fixed ε > 0. We point here that this hardness result applies only to graphs with large chromatic numbers. The second result shows that for any positive constant h, there exists an integer , such that it is NP-hard to decide whether a given graph G is -chromatic or any coloring of G requires colors. Received April 11, 1997/Revised June 10, 1999     

Applying a proof of tverberg to complete bipartite decompositions of digraphs and multigraphs

Graham and Pollak [2] proved that n – 1 is the minimum number of edge-disjoint complete bipartite subgraphs into which the edges of K_n decompose. Tverberg [6], using a linear algebraic technique, was the first to give a simple proof of this result. We apply Tverberg's technique to obtain results for two related decomposition problems, in which we wish to partition the arcs/edges of complete digraphs/multigraphs into a minimum number of arc/edge-disjoint complete bipartite subgraphs of appropriate natures. We obtain exact results for the digraph problem, which was posed by Lundgren and Maybee [4]. We also obtain exact results for the decomposition of complete multigraphs with exactly two edges connecting every pair of distinct vertices.     

Planar graph coloring with an uncooperative partner

We show that the game chromatic number of a planar graph is at most 33. More generally, there exists a function f: ? → ? so that for each n ∈ ?, if a graph does not contain a homeomorph of K_n, then its game chromatic number is at most f(n). In particular, the game chromatic number of a graph is bounded in terms of its genus. Our proof is motivated by the concept of p-arrangeability, which was first introduced by Guantao and Schelp in a Ramsey theoretic setting.     

Relations between the Local Chromatic Number and Its Directed Version

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm.     

List coloring digraphs

The dichromatic number of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann‐Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this article, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as . We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.     

Total chromatic number of complete r-partite graphs

Rosenfeld (1971) proved that the Total Colouring Conjecture holds for balanced complete r-partite graphs. Bermond (1974) determined the exact total chromatic number of every balanced complete r-partite graph. Rosenfeld's result had been generalized recently to complete r-partite graphs by Yap (1989). The main result of this paper is to prove that the total chromatic number of every complete r-partite graph G of odd order is Δ (G) + 1. This result gives a partial generalization of Bermond's theorem.