A self-stabilizing algorithm for coloring planar graphs

Abstract

Summary. This paper describes an algorithm for coloring the nodes of a planar graph with no more than six colors using a self-stabilizing approach. The first part illustrates the coloring algorithm on a directed acyclic version of the given planar graph. The second part describes a self-

Transformations for Write-All-with-Collision Model

“…We show that previously studied concepts such as graph coloring (e.g., [22][23][24]), local mutual exclusion (e.g., [19,20]) and time division multiple access (TDMA) [25,26] can be effectively used for obtaining these transformations. The main contributions of the paper are as follows:…”

Fault tolerant algorithms for orderings and colorings

“…A well-known theorem of Brooks [12] says that any for any graph G, χ(G) ≤ ∆(G) + 1, where ∆(G) denotes the maximum degree of a node in G. S-s algorithms which find (∆ + 1)-colorings, under various models, are presented by Dolev and Herman [6], Gradinariu and Tixeuil [10], and Hedetniemi et al [11]. Ghosh and Karaata in [9] give an s-s algorithm to find a 6-coloring of a planar graph, using the fact that every planar graph has a node of degree less than 6.…”

“…Special cases of this algorithm 6-color any planar graph, thus generalizing an s-s algorithm of Ghosh and Karaata [9], 2-color any tree and 3-color any series-parallel graph.…”

“…The first algorithm, which we call Leap, is the natural generalization of an algorithm by Ghosh and Karaata [9]. However, we show that, while it is efficient for k = 2, their algorithm can take exponentially many moves.…”

“…The algorithm has a single rule: if a node v has at least k neighbors having the same number or larger number, it sets w(v) to be one larger than the maximum number of a neighbor. This is the natural generalization an s-s algorithm of Ghosh and Karaata [9], while avoiding the use of IDs. The fact that IDs are unnecessary was observed by Shukla et al in [14].…”