Exact values of defective Ramsey numbers in graph classes

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

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A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽

10.3390/a14060164 ◽

2021 ◽

Vol 14 (6) ◽

pp. 164

Author(s):

Tobias Rupp ◽

Stefan Funke

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Real World ◽

Road Network ◽

Upper And Lower Bounds ◽

Bounded Degree ◽

Shortest Path Routing ◽

Graph Classes ◽

Speed Up ◽

To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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A Note on On-Line Ramsey Numbers of Stars and Paths

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-021-01130-x ◽

2021 ◽

Author(s):

Fatin Nur Nadia Binti Mohd Latip ◽

Ta Sheng Tan

Keyword(s):

Ramsey Numbers ◽

On Line

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Bipartite Ramsey numbers of Kt,s in many colors

Applied Mathematics and Computation ◽

10.1016/j.amc.2021.126220 ◽

2021 ◽

Vol 404 ◽

pp. 126220

Author(s):

Ye Wang ◽

Yusheng Li ◽

Yan Li

Keyword(s):

Ramsey Numbers

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Isomorphism, canonization, and definability for graphs of bounded rank width

Communications of the ACM ◽

10.1145/3453943 ◽

2021 ◽

Vol 64 (5) ◽

pp. 98-105

Author(s):

Martin Grohe ◽

Daniel Neuen

Keyword(s):

Fixed Point ◽

Complexity Theory ◽

Polynomial Time ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Descriptive Complexity ◽

Graph Isomorphism Problem ◽

Structural Graph ◽

Graph Classes ◽

Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

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Turán theorems for unavoidable patterns

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500412100027x ◽

2021 ◽

pp. 1-20

Author(s):

ANTÓNIO GIRÃO ◽

BHARGAV NARAYANAN

Keyword(s):

Complete Graph ◽

Blow Up ◽

Ramsey Numbers ◽

Order Of Magnitude ◽

Transitive Tournament ◽

Unavoidable Patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

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