On colorings of squares of outerplanar graphs

Engineering Research Institute, University of Iceland, Hjarðarhagi 2-6, IS-107 Reykjavík, Iceland
On Colorings of Squares of Outerplanar Graphs∗
Geir Agnarsson
The results or opinions presented in this report are the responsibility of the authors. They should not be interpreted as representing the Research Institute or the University of Iceland. © Engineering Research Institute, University of Iceland, and the authors
†
Magn´ us M. Halld´ orsson
‡
Abstract We study vertex colorings of the square G2 of an outerplanar graph G. We find the optimal bound of the inductiveness, chromatic number and the clique number of G2 as a function of the maximum degree ∆ of G for all ∆ ∈ N. As a bonus, we obtain the optimal bound of the choosability (or the list-chromatic number) of G2 when ∆ ≥ 7. In the case of chordal outerplanar graphs, we classify exactly which graphs have parameters exceeding the absolute minimum. 2000 MSC: 05C05, 05C12, 05C15. Keywords: outerplanar, chordal, weak dual, power of a graph, greedy coloring, chromatic number, clique number, inductiveness.
An earlier version of this current paper appeared in SODA 2004 [2]. Department of Mathematical Sciences, George Mason University, MS 3F2, 4400 University Drive, Fairfax, VA 22030, geir@ ‡ Department of Computer Science, University of Iceland, Reykjav´ ık, Iceland. mmh@hi.is
Geir Agnarsson, Magnús M. Halldórsson. On Coloring Squares of Outerplanar Graphs, Engineering Research Institute, University of Iceland, Technical report VHI-04-2005, December 2005
On Coloring Squares of Outerplanar Graphs
Geir Agnarsson Magnús M. Halldórsson
December 2005
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On Coloring Squares of Outerplanar Graphs
Report nr. VHI-04-2005, Reykjavík 2005
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in the size of the graph [7]. The algorithm has also the special advantage that it requires only the square graph G2 and does not require information about the underlying graph G. The purpose of this article is to further contribute to the study of various vertex colorings of squares of planar graphs, by examining an important subclass of them, the class of outerplanar graphs. Observe that the neighborhood of a vertex with ∆ neighbors induces a clique in the square graph. Thus, the chromatic number, and in fact the clique number, of any graph of maximum degree ∆ is necessarily a function of ∆ and always at least ∆ + 1. Our results. We derive tight bounds on chromatic number, as well as the inductiveness and the clique number of the square of an outerplanar graph G as a function of the maximum degree ∆ of G. One of the main results, given in Section 3, is that when ∆ ≥ 7, the inductiveness of G 2 is exactly ∆. It follows that the clique and chromatic numbers are exactly ∆ + 1 and that Greedy yields an optimal coloring. As a bonus we obtain in this case that the choosability (see Definition 3.11) is the optimal ∆ + 1. We can then treat the low-degree cases separately to derive a linear-time algorithm independent of ∆. We examine in detail the low-degree cases, ∆ < 7, and derive best possible upper bounds on the maximum clique and chromatic numbers, as well as inductiveness of squares of outerplanar graphs. These bounds are illustrated in Table 1. We treat the special case of chordal outerplanar graphs separately, and further classify all chordal outerplanar graphs G for which the inductiveness of G2 exceeds ∆ or the clique or chromatic number of G 2 exceed ∆ + 1. ∆ 2 3 4 5 6 7+ ω ∆+1 ∆+1 ∆+2 ∆+1 ∆+1 ∆+1 Chordal ind ∆ ∆ ∆+1 ∆+1 ∆+1 ∆ χ ∆+1 ∆+1 ∆+2 ∆+1 ∆+1 ∆+1 ω ∆+3 ∆+2 ∆+2 ∆+1 ∆+1 ∆+1 General ind ∆+2 ∆+1 ∆+2 ∆+1 ∆+1 ∆ χ ∆+3 ∆+2 ∆+2 ∆+2 ∆+1 ∆+1
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Introduction
The square of a graph G is the graph G2 on the same vertex set with edges between pair of vertices of distance one or two in G. Coloring squares of graphs has been studied, e.g., in relation to frequency allocation. This models the case when nodes represent both senders and receivers, and two senders with a common neighbor will interfere if using the same frequency. The problem of coloring squares of graphs has particularly seen much attention on planar graphs. A conjecture of Wegner [12] dating from 1977 (see [8]), states that the square of every planar graph G of maximum degree ∆ ≥ 8 has a chromatic number which does not exceed 3∆/2 + 1. The conjecture matches the maximum clique number of these graphs. Currently the best upper bound known is 1.66∆ + 78 by Molloy and Salavatipour [11]. An earlier paper of the current authors [1] gave a bound of 1.8∆ for the chromatic number of squares of planar graph with large maximum degree ∆ ≥ 749. This is based on bounding the inductiveness of the graph, which is the maximum over all subgraphs H of the minimum degree of H . It was also shown there that this was the best possible bound on the inductiveness. Borodin et al [4] showed that this bound holds for all ∆ ≥ 48. Inductiveness has the additional advantage of also bounding the list-chromatic number. Inductiveness leads to a natural greedy algorithm (henceforth called Greedy): Select vertex u ∈ V (G) of minimum degree, sometimes called a simplicial vertex of G, recursively color G \ u, and finally color u with the smallest available color. Alternatively, k -inductiveness leads to an inductive ordering u1 , u2 , . . . , un of the vertices such that any vertex u i has at most k neighbors among {ui+1 , . . . , un }. Then, if we color the vertices first-fit in the reverse order u n , un−1 , . . . , u1 (i.e. assigning each vertex the smallest color not used among its previously colored neighbors), the number of colors used is at most k + 1. Implemented efficiently, the algorithm runs in time linear