Every planar map is four colorable. Part II: Reducibility

Appel, K. I.; Haken, Wolfgang; Koch, Jean E.

doi:10.1215/ijm/1256049012

Cited by 616 publications

(479 citation statements)

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“…Theorem 1.4 is a partial result for Conjectures 1.2 and 1.3. By applying the 4-color theorem ( [1], [2], [3] and [7]), we also have the following corollary, which is a strengthening of the results of Hind ( [12]) and Goldwasser and Zhang ([10]) mentioned above because it applies to every counterexample, not just a minimum counterexample.…”

Section: Conjecture 11 (Fiorini and Wilsonsupporting

confidence: 54%

Uniquely Edge-3-Colorable Graphs and Snarks

Goldwasser

Zhang

2000

Graphs Comb

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A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edgeconnected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G nT is acyclic and a cyclic edge-cut T is trivial if one component of G nT is a circuit of length jTj).

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Section: Conjecture 11 (Fiorini and Wilsonsupporting

confidence: 54%

Uniquely Edge-3-Colorable Graphs and Snarks

Goldwasser

Zhang

2000

Graphs Comb

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“…χ(G) = 2 if and only if G is bipartite and this can be tested in linear time. The chromatic number of a planar graph is at most four [2,3], but its computation is N P-hard [28]. Also o1p graphs may need four colors for their K 4 s. However, for o1p graphs the chromatic number can be computed in linear time since o1p graphs have bounded treewidth [4].…”

Section: Theoremmentioning

confidence: 99%

Outer 1-Planar Graphs

et al. 2015

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A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is N P-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph G is o1p, then the algorithm computes an embedding and can augment G to a maximal o1p graph. Otherwise, G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) Grant Br835/18-1. B Christian Bachmaier

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“…It follows from Theorem 3(c), (d) that P (G, 1) = 0 and that P (G, t) becomes negative when t increases from 1. If, in addition, we assume that G is bipartite, then P (G, 2) = 2 and hence P (G, t) must have a zero in (1,2). We may combine these graphs with any other graph using Lemma 2, with r = 2, to construct 2-connected non-bipartite graphs with chromatic roots in (1,2).…”

Section: -Connected Graphsmentioning

confidence: 99%

“…Their hope was that results from analysis and algebra could be used to prove their stronger conjecture and hence deduce that the 4-colour conjecture was true. This has not yet occurred: indeed the 4-colour conjecture is now a theorem [1,2,23], but the stronger conjecture of Birkhoff and Lewis remains unsolved. Nevertheless, many beautiful results have been obtained concerning the zero distribution of chromatic polynomials both on the real line and in the complex plane, and many other intriguing problems remain open.…”

Section: Introductionmentioning

confidence: 99%