
On the Graceful Game
A graceful labeling of a graph G with m edges consists of labeling the v...
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A Connected Version of the Graph Coloring Game
The graph coloring game is a twoplayer game in which, given a graph G a...
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Induced universal graphs for families of small graphs
For 0 ≤ k ≤ 6, we give the minimum number of vertices f(k) in a graph co...
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Eternal kdomination on graphs
Eternal domination is a dynamic process by which a graph is protected fr...
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Almostmonochromatic sets and the chromatic number of the plane
In a colouring of R^d a pair (S,s_0) with S⊆R^d and with s_0∈ S is almos...
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Graph pattern detection: Hardness for all induced patterns and faster noninduced cycles
We consider the pattern detection problem in graphs: given a constant si...
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The localization capture time of a graph
The localization game is a pursuitevasion game analogous to Cops and Ro...
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On Induced Online Ramsey Number of Paths, Cycles, and Trees
An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a graph H and a graph G of an infinite set of independent vertices. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some finite round there is a monochromatic copy of the graph H, otherwise Painter wins. The online Ramsey number r(H) is the minimum number of rounds such that Builder can force a monochromatic copy of H in G. This is an analogy to the sizeRamsey number r(H) defined as the minimum number such that there exists graph G with r(H) edges where for any edge twocoloring G contains a monochromatic copy of H. In this paper, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number r_ind(H) is the minimum number of rounds Builder can force an induced monochromatic copy of H in G. We prove asymptotically tight bounds on the induced online Ramsey numbers of paths, cycles and two families of trees. Moreover, we provide a result analogous to Conlon [Online Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees T_1,T_2,..., T_i<T_i+1 for i>1, such that _i→∞r(T_i)/r(T_i) = 0.
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