Israel R. Curbelo

Sam Beilis, Israel R. Curbelo

The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. Each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy that guarantees a red copy of $G$ or a blue copy of $H$ in at most $t$ rounds. We show that for every fixed $k$, there are constants $λ_1$ and $λ_2$ such that $\tilde{r}(P_k,P_n)/n$ and $\tilde{r}(P_k,C_n)/n$ converge to $λ_1$, and $\tilde{r}(K_{1,k},P_n)/n$ and $\tilde{r}(K_{1,k},C_n)/n$ converge to $λ_2$.

Israel R. Curbelo

We study the online coloring of $σ$-interval graphs, which are interval graphs with interval lengths in $[1,σ]$ and 2-count interval graphs, which are interval graphs that require at most two distinct interval lengths. For $σ$-interval graphs, the Kierstead-Trotter algorithm has competitive ratio 3 and no online algorithm has competitive ratio better than 2. In this paper, we show that for every $ε>0$, there is a $σ>1$ such that there is no online algorithm for $σ$-interval coloring with competitive ratio less than $3-ε$. For 2-count interval graphs, we show that the greedy algorithm First-Fit has competitive ratio at most $4$, that there is no online algorithm with competitive ratio less than $2.5$ when the interval representation is unknown, and that there is no online algorithm with competitive ratio less than $2$ when the interval representation is known