On Induced Online Ramsey Number of Paths, Cycles, and Trees

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F19%3A00333848" target="_blank" >RIV/68407700:21240/19:00333848 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/chapter/10.1007/978-3-030-19955-5_6" target="_blank" >https://link.springer.com/chapter/10.1007/978-3-030-19955-5_6</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-030-19955-5_6" target="_blank" >10.1007/978-3-030-19955-5_6</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On Induced Online Ramsey Number of Paths, Cycles, and Trees

Popis výsledku v původním jazyce

An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a fixed graph $H$ and a an infinite set of independent vertices $G$. In each round Builder draws a new edge in $G$ 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 $widetilde{r}(H)$ is the minimum number of rounds such that Builder can force a monochromatic copy of $H$ in $G$. This is an analogy to the size-Ramsey number $overline{r}(H)$ defined as the minimum number such that there exists graph $G$ with $overline{r}(H)$ edges where for any edge two-coloring $G$ contains a monochromatic copy of $H$. In this extended abstract, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number $overline{r}_{ind}(H)$ 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 [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees $T_1,T_2,dots$, $|T_i|<|T_{i+1}|$ for $ige1$, such that [ lim_{itoinfty} frac{widetilde{r}(T_i)}{overline{r}(T_i)} = 0. ]

Název v anglickém jazyce

On Induced Online Ramsey Number of Paths, Cycles, and Trees

Popis výsledku anglicky

An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a fixed graph $H$ and a an infinite set of independent vertices $G$. In each round Builder draws a new edge in $G$ 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 $widetilde{r}(H)$ is the minimum number of rounds such that Builder can force a monochromatic copy of $H$ in $G$. This is an analogy to the size-Ramsey number $overline{r}(H)$ defined as the minimum number such that there exists graph $G$ with $overline{r}(H)$ edges where for any edge two-coloring $G$ contains a monochromatic copy of $H$. In this extended abstract, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number $overline{r}_{ind}(H)$ 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 [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees $T_1,T_2,dots$, $|T_i|<|T_{i+1}|$ for $ige1$, such that [ lim_{itoinfty} frac{widetilde{r}(T_i)}{overline{r}(T_i)} = 0. ]

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

The 14th International Computer Science Symposium in Russia

ISBN

978-3-030-19954-8

ISSN

0302-9743

e-ISSN

—

Počet stran výsledku

10

Strana od-do

60-69

Název nakladatele

Springer, Cham

Místo vydání

—

Místo konání akce

Novosibirsk

Datum konání akce

1. 7. 2019

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

000490894900006