# Linear Bounds for Cycle-free Saturation Games

@article{English2022LinearBF, title={Linear Bounds for Cycle-free Saturation Games}, author={Sean English and Tom'avs Masavr'ik and Grace McCourt and Erin Meger and Michael S. Ross and Sam Spiro}, journal={ArXiv}, year={2022}, volume={abs/2108.05295} }

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph in $\mathcal{F}$. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let $\textrm{sat}_g(n,\mathcal{F…

## One Citation

### Project Description Sam Spiro

- Mathematics
- 2021

Broadly speaking, questions in extremal combinatorics ask how large or small a combinatorial object can be. For example, a classical theorem of Mantel’s [20] states that every n-vertex triangle-free…

## References

SHOWING 1-10 OF 14 REFERENCES

### Saturation Games for Odd Cycles

- MathematicsElectron. J. Comb.
- 2019

It is shown that the number of edges that are in the final graph when both players play optimally is the same, andsat_g(\mathcal{C}_\infty\setminus C_3;n) is proved to be 2n-2, where $k\ge 2$ denotes the set of all odd cycles.

### Colorability Saturation Games

- MathematicsArXiv
- 2016

Almost matching lower and upper bounds on the score of the game that hold for arbitrary choices of $k$ and $n>k$ are provided and it is proved that its score is $n^2/3+O(n)$.

### Saturation Numbers for Trees

- MathematicsElectron. J. Comb.
- 2009

The saturation numbers of ${\bf sat}(n,T_k)$ for some families of trees will be determined precisely, and the maximum and minimum values of the saturation number for the class of all Trees will be given.

### Game saturation of intersecting families

- Mathematics
- 2012

We consider the following combinatorial game: two players, Fast and Slow, claim k-element subsets of [n] = {1, 2, …, n} alternately, one at each turn, so that both players are allowed to pick sets…

### The Game Saturation Number of a Graph

- MathematicsJ. Graph Theory
- 2017

The lower bound on the length under optimal play of the P_4-saturation game on K_{m,n} is proved, which is similar to the results when Min plays first.

### An upper bound on the extremal version of Hajnal's triangle-free game

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2016

### ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS

- Mathematics
- 2002

In 1940 TURIN raised the following question: if the number of nodes, n, of a graph’ is prescribed and if I is an integer cs n, what is the number of edges which the graph has to contain in order to…

### Graduate Texts in Mathematics

- Education
- 1982

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully…