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The Two-Coloring Number and Degenerate Colorings of Planar Graphs∗

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The Two-Coloring Number and Degenerate Colorings of Planar Graphs∗ SIAM J. DISCRETE MATH. c 2009 Society for Industrial and Applied Mathematics Vol. 0, No. 0, pp. 000–000 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS OF PLANAR GRAPHS∗ HAL KIERSTEAD†, BOJAN MOHAR‡,SIMONSPACAPANˇ §, DAQING YANG¶, AND XUDING ZHU Abstract. The two-coloring number of graphs, which was originally introduced in the study of the game chromatic number, also gives an upper bound on the degenerate chromatic number as introduced by Borodin. It is proved that the two-coloring number of any planar graph is at most nine. As a consequence, the degenerate list chromatic number of any planar graph is at most nine. It is also shown that the degenerate diagonal chromatic number is at most 11 and the degenerate diagonal list chromatic number is at most 12 for all planar graphs. Key words. two-coloring number, degenerate coloring, planar graph AMS subject classification. 05C15 DOI. 10.1137/070703697 1. Introduction. The two-coloring number of a graph was introduced by Chen and Schelp [6] in the study of Ramsey properties of graphs and used later in the study of the game chromatic number [9, 10, 15, 16]. In [10] it was shown that the two-coloring number is related to the acyclic chromatic number. It turns out that the two-coloring number is also related to the notion of degenerate colorings introduced by Borodin [1], which was the starting motivation for the results of this paper. Let G be a graph, and let L be a linear ordering of V (G). A vertex x is k-reachable from y if x<L y and there is an xy-path P of length at most k such that y<L z for all interior vertices z of P .LetRL,k(y) be the set of all vertices x that are k-reachable from y with respect to the linear order L.Thek-coloring number of G is defined as colk(G)=1+min max |RL,k(y)|, L y∈V (G) where the minimum is taken over all linear orderings L of V (G). If k =1,then col1(G)isalsoknownasthecoloring number of G since it provides an upper bound ∗Received by the editors September 25, 2007; accepted for publication (in revised form) July 7, 2009; published electronically DATE. http://www.siam.org/journals/sidma/x-x/70369.html †Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804 ([email protected]). This author’s research was supported in part by NSA grant H98230-08-1-0069. ‡Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada ([email protected]). On leave from Department of Mathematics, IMFM & FMF, University of Ljubljana, Ljubljana, Slovenia. This author’s research was supported in part by the ARRS (Slovenia), Research Program P1–0297, by an NSERC Discovery grant, and by the Canada Research Chair program. §University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia (simon.spacapan@uni- mb.si). This author’s research was supported in part by the ARRS (Slovenia), Research Program P1–0297. ¶Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian 350002, China (daqing85@ yahoo.com). This author’s research was supported in part by NSFC under grants 10771035 and SX2006-42 of the college of Fujian. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Tai- wan, People’s Republic of China, and National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan, People’s Republic of China ([email protected]). This au- thor’s research was supported by Taiwan grant NSC95-2115-M-110-013-MY3. 1 2 KIERSTEAD, MOHAR, SPACAPAN,ˇ YANG, AND ZHU for the chromatic number of G. The higher k-coloring numbers provide upper bounds for some other coloring parameters [17]. Let k be a positive integer. A graph G is k-degenerate if every subgraph of G has a vertex of degree less than k. A coloring of a graph such that for every k ≥ 1, the union of any k color classes induces a k-degenerate subgraph, is a degenerate coloring. Note that this strengthens the notion of acyclic colorings, for which it is required that every color class is 1-degenerate and the union of any two color classes induces a 2-degenerate graph (a forest). The degenerate chromatic number of G, denoted as χd(G), is the least n such that G admits a degenerate coloring with n colors. Suppose that for each vertex v ∈ V (G) we assign a list L(v) ⊂ N of admissible colors which can be used to color the vertex v.Alist coloring of G is a function f : V (G) → N, such that f(v) ∈ L(v)foreachv ∈ V (G)andf(u) = f(v) whenever u and v are adjacent in G. If the coloring is also degenerate, we say that f is a degenerate list coloring. If for any choice of lists L(v),v ∈ V (G), such that |L(v)|≥k,thereexistsa list coloring of G,thenwesaythatG is k-choosable.Thelist chromatic number (or the choice number)ofG, denoted as ch(G), is the least k, such that G is k-choosable. Analogously we define the degenerate choice number and denote it by chd(G). In 1976 Borodin proved [2, 3] that every planar graph admits an acyclic 5-coloring and thus solved a conjecture proposed by Gr¨unbaum [7]. At the same time, he pro- posed the following conjecture. Conjecture 1 (Borodin [2, 3]). Every planar graph has a 5-coloring such that the union of every k color classes with 1 ≤ k ≤ 4 induces a k-degenerate graph. Thomassen settled a weakening of Conjecture 1 by proving that the vertex set of every planar graph can be decomposed into two sets that, respectively, induce a 2-degenerate graph and a 3-degenerate graph [13], and that the vertex set of every planar graph can be decomposed into an independent set and a set that induces a 4-degenerate graph [14]. However, Conjecture 1 remained basically untouched since there are no tools to deal with degenerate colorings. Very recently, the barrier was overriden by Rautenbach [11] who proved that every planar graph admits a degenerate coloring using at most 18 colors. In this paper we introduce two different approaches for dealing with degenerate colorings. Both are based on the following observation. Observation 1. Let G be a graph, and let c be a degenerate coloring of a vertex- deleted subgraph G − v. If the neighbors of v are colored by pairwise distinct colors and we color v by a color which is different from all of those colors, then the resulting coloring of G is degenerate. The observation above, whose easy proof is left to the reader, is a link between the two-coloring number and the degenerate chromatic number, since it implies the following proposition. Proposition 1. For any graph G, chd(G) ≤ col2(G). Proof. Suppose that L is a linear ordering of V (G), and suppose that each vertex y has a list of |RL,2(y)| + 1 colors. Then we can color the vertices of G,onebyone, according to the linear ordering L,sothateachvertexy is colored by a color different from the colors of vertices in RL,2(y). This strategy guarantees that all neighbors of y, which are already colored, have pairwise different colors. To see this, let y1 <L y2 be any two neighbors of y,wherey1,y2 <L y.Theny1 ∈ RL,2(y2), so y2 has been colored differently from y1. It follows from Observation 1 that the obtained coloring of G is degenerate and hence chd(G) ≤ col2(G). In this paper we shall use two-colorings to get an efficient bound on the degenerate THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 3 chromatic number. It is proved in [9] that planar graphs have a two-coloring number at most 10. Currently, our main result gives the best upper bound for the two-coloring number of every planar graph G, and henceforth a bound on chd(G). Theorem 1. If G is a planar graph, then col2(G) ≤ 9, and therefore also chd(G) ≤ 9. Theorem 1 will be proved in section 2. In fact, we will prove a slightly stronger statement that there is a linear ordering L of V (G) such that for each vertex x, |RL,1(x)|≤5and|RL,2(x)|≤8. See Theorem 2. An example given in [9] shows that there are planar graphs whose 2-coloring number is 8. That example is quite complicated. Here we give a much simpler one. Consider a five-connected triangulation T of the plane in which no two vertices of degree 5 are adjacent. It is well known and easy to see that there are infinitely many such triangulations. Let L be a linear ordering of vertices of T ,andletx be the last vertex that has a larger neighbor with respect to L. Suppose that y1,...,yk are − the neighbors of x such that x<L yi for i =1,...,k,andletN (x)bethesetof neighbors of x distinct from y1,...,yk. By the choice of x, all neighbors of y1,...,yk − are 2-reachable from x. Note that each yi hasatmosttwoneighborsinN (x). If − k = 1, it is easy to see that N (x) and the neighbors of y1 contain at least seven vertices distinct from x.Thus,|RL,2(x)|≥7. On the other hand, if k ≥ 2, then y1 and y2 have only x and possibly one neighbor of x as common neighbors. If one of them has degree at least 6, then they have at least seven neighbors distinct from x, and hence |RL,2(x)|≥7.
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