Let's say that a graph family $\mathcal{F}$ has long induced paths if there is a constant $\epsilon > 0$ such that every graph $G$ in $\mathcal{F}$ contains an induced path on $|V(G)|^{\epsilon}$ vertices. I am interested in properties of graph families that ensure the existence of long induced paths. In particular, I am currently wondering whether constant-degree expanders have long induced paths. Here is what I know.

• Random graphs with constant average degree (in the Erdős–Rényi model) have long (even linear-size) induced paths with high probability; see for example Suen's article.
• Unique-neighbor expander graphs (as defined by Alon and Copalbo) have long induced paths. In fact, any maximal induced path is long in such graphs.

Given these two facts I would expect that contant-degree expanders have long induced paths. However, I was unable to find any concrete results. Any insights are much appreciated.

Let's say that a graph family $\mathcal{F}$ has long induced paths if there is a constant $\epsilon > 0$ such that every graph $G$ in $\mathcal{F}$ contains an induced path on $|V(G)|^{\epsilon}$ vertices. I am interested in properties of graph families that ensure the existence of long induced paths. In particular, I am currently wondering whether constant-degree expanders have long induced paths. Here is what I know.

• Random graphs with constant average degree (in the Erdős–Rényi model) have long (even linear-size) induced paths with high probability; see for example Suen's article.
• Unique-neighbor expander graphs (as defined by Alon and Copalbo) have large induced trees. In fact, any maximal induced tree is large in such graphs.

Given these two facts I would expect that contant-degree expanders have long induced paths. However, I was unable to find any concrete results. Any insights are much appreciated.