Coloring Triangle-Free Graphs on Surfaces

Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin

doi:10.1007/978-3-540-77120-3_2

Cited by 21 publications

(56 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…Nevertheless, in [7] we give a polynomialtime algorithm to test whether a drawing in a fixed surface with no contractible triangles is 3-colorable. In particular, this algorithm applies to even-faced drawings in the Klein bottle.…”

Section: Discussionmentioning

confidence: 99%

“…When Σ is the sphere this is, of course, trivial by Theorem 1, and when Σ is the projective plane a polynomialtime algorithm follows from Theorem 4 below. Recently, we were able to find [7] a polynomial-time algorithm for any fixed surface Σ, but when Σ has non-positive Euler characteristic there does not seem to be a simple characterization of 3-colorability.…”

Section: Theorem 1 Every Triangle-free Planar Graph Is 3-colorablementioning

confidence: 99%

Coloring even-faced graphs in the torus and the Klein bottle

Kráľ

Thomas

2008

Combinatorica

Self Cite

View full text Add to dashboard Cite

We prove that a triangle-free graph drawn in the torus with all faces bounded by even walks is 3-colorable if and only if it has no subgraph isomorphic to the Cayley graph C(Z13; 1, 5). We also prove that a non-bipartite quadrangulation of the Klein bottle is 3-colorable if and only if it has no non-contractible separating cycle of length at most four and no odd walk homotopic to a non-contractible two-sided simple closed curve. These results settle a conjecture of Thomassen and two conjectures of Archdeacon, Hutchinson, Nakamoto, Negami and Ota.

show abstract

“…The most recent results for determining the 3-colorability of a graph are based on recognizing if the input graph is planar and triangle-free [11], [12], or discovering some relationship between the topology of the graph and its structural parts [6]- [8], [13]- [17].…”

Section: The Graph Vertex Coloring Problemmentioning

confidence: 99%

“…Gimbel and Thomasson [17] found an elegant 3-colorability proof for triangle-free projective planar graphs. Dvorá k [13] found a linear-time algorithm to decide whether a triangle-free graph in a general surface Σ is 3-colorable. 3-colorability is also polynomially solvable for graphs containing no induced path on 5 vertices [15].…”

Section: The Graph Vertex Coloring Problemmentioning

confidence: 99%

A Note for the Two Incremental Satisﬁability Problem

R.¹,

L.²,

L.³

2017

IJCTE

View full text Add to dashboard Cite

Abstract-Let K be a two conjunctive normal form and φ a three conjunctive normal form, both formulas are defined over the same set of variables. It is well known that SAT(K) is in the complexity class P, while SAT(φ) is a classic NP-Complete problem. We consider the computational complexity of determining SAT(K φ ) as an incremental satisfiability problem (2-ISAT). We show that this problem is NP-complete even if the number of occurrences of each variable in φ is one.Also, we propose a method to review SAT(Kφ ). Our proposal is adequate to solve 2-ISAT problem. Our algorithm allows us to recognize tractable instances of 2-ISAT.

show abstract

Coloring Triangle-Free Graphs on Surfaces

Cited by 21 publications

References 16 publications

Deciding First-Order Properties for Sparse Graphs

Deciding First-Order Properties for Sparse Graphs

Coloring even-faced graphs in the torus and the Klein bottle

A Note for the Two Incremental Satisﬁability Problem

Contact Info

Product

Resources

About