# tree

## A gold-grabbing game ★★

Author(s): Rosenfeld

** Setup** Fix a tree and for every vertex a non-negative integer which we think of as the amount of *gold* at .

**2-Player game** Players alternate turns. On each turn, a player chooses a leaf vertex of the tree, takes the gold at this vertex, and then deletes . The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.

**Problem**Find optimal strategies for the players.

## Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family of graphs is -*bounded* if there exists a function so that every satisfies .

**Conjecture**For every fixed tree , the family of graphs with no induced subgraph isomorphic to is -bounded.

Keywords: chi-bounded; coloring; excluded subgraph; tree

## Does the chromatic symmetric function distinguish between trees? ★★

Author(s): Stanley

**Problem**Do there exist non-isomorphic trees which have the same chromatic symmetric function?

Keywords: chromatic polynomial; symmetric function; tree

## Graham's conjecture on tree reconstruction ★★

Author(s): Graham

**Problem**for every graph , we let denote the line graph of . Given that is a tree, can we determine it from the integer sequence ?

Keywords: reconstruction; tree