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Keywords = on-line Ramsey number

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7 pages, 266 KB  
Article
Online and Connected Online Ramsey Numbers of a Matching versus a Path
by Ruyu Song and Yanbo Zhang
Symmetry 2022, 14(11), 2277; https://doi.org/10.3390/sym14112277 - 31 Oct 2022
Cited by 1 | Viewed by 1867
Abstract
The (G1,G2)-online Ramsey game is a two-player turn-based game between a builder and a painter. Starting from an empty graph with infinite vertices, the builder adds a new edge in each round, and the painter colors [...] Read more.
The (G1,G2)-online Ramsey game is a two-player turn-based game between a builder and a painter. Starting from an empty graph with infinite vertices, the builder adds a new edge in each round, and the painter colors it red or blue. The builder aims to force either a red copy of G1 or a blue copy of G2 in as few rounds as possible, while the painter’s aim is the opposite. The online Ramsey number r˜(G1,G2) is the minimum number of edges that the builder needs to win the (G1,G2)-online Ramsey game, regardless of the painter’s strategy. Furthermore, we initiate the study of connected online Ramsey game, which is identical to the usual one, except that at any time the graph induced by all edges should be connected. In this paper, we show a general bound of the online Ramsey number of a matching versus a path and determine its exact value when the path has an order of three or four. For the connected version, we obtain all connected online Ramsey numbers of a matching versus a path. Full article
(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)
6 pages, 249 KB  
Article
A Note on On-Line Ramsey Numbers for Some Paths
by Tomasz Dzido and Renata Zakrzewska
Mathematics 2021, 9(7), 735; https://doi.org/10.3390/math9070735 - 29 Mar 2021
Cited by 2 | Viewed by 2617
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematical Modelling and Multi-Criteria Optimisation in Engineering)
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