Three-coloring triangle-free planar graphs in linear time

Dvořák, Zdeněk; Kawarabayashi, Ken-ichi; Thomas, Robin

doi:10.1145/2000807.2000809

Cited by 33 publications

(54 citation statements)

References 19 publications

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“…Since t 3 has degree at least three, this chord exists and joins t 3 with r 5 , and hence G has a subgraph isomorphic to the graph depicted in Figure 1(b), contrary to (8). This proves (11). (12) G does not contain a 5-face incident only with internal vertices of degree three.…”

Section: (11)mentioning

confidence: 84%

“…Let L ′ be the triangle of G ′′ . By (11) we have L ′ = T , and hence x 1 x 3 is an edge of L ′ . Let t be the third vertex of L ′ .…”

Section: (11)mentioning

confidence: 99%

“…Recently, two of us, in joint work with Kawarabayashi [11] were able to design a linear-time algorithm to 3-color triangle-free planar graphs, and as a by-product found perhaps a yet simpler proof of Theorem 2. The statement of Theorem 2 cannot be extended to any surface other than the sphere.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle

Dvořák

Kráľ

Thomas

2016

Journal of Combinatorial Theory, Series B

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Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.

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Section: (11)mentioning

confidence: 84%

“…Let L ′ be the triangle of G ′′ . By (11) we have L ′ = T , and hence x 1 x 3 is an edge of L ′ . Let t be the third vertex of L ′ .…”

Section: (11)mentioning

confidence: 99%

See 1 more Smart Citation

Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle

Dvořák

Kráľ

Thomas

2016

Journal of Combinatorial Theory, Series B

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“…The second of these data structures is needed in our linear-time algorithm for 3-coloring triangle-free graphs on surfaces [7], also see [6]. The last one is inspired by a data structure of Kowalik and Kurowski [15], [17] for deciding whether two vertices of a planar graph are connected by a path of length at most k, where k is a fixed constant.…”

Section: B Our Resultsmentioning

confidence: 99%

Deciding First-Order Properties for Sparse Graphs

Dvořák

Kráľ

Thomas

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

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Abstract-We present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost linear-time algorithm for deciding FOL properties in classes of graphs with locally bounded expansion; those include classes of graphs with locally bounded tree-width or locally excluding a minor.More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings.

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“…If the search yields no triangles in P , then by [Grö59] such a P is 3-colorable, and so we map P to a fixed list with a consistent mapping, say L = L 1 = {a, b, c}. (In fact, by [DKT11] it is known that triangle-free planar graphs can be colored in linear time. )…”

Section: Lemma 6 Cmp Is Np-hardmentioning

confidence: 99%