Linear host orders for trees

Establish whether there is an absolute constant c such that every tree T on n vertices satisfies η(T)≤cn.

Background

The paper proves the universal upper bound η(T)≤n2 for trees but has no linear bound. Its computations establish η(P_n)=n+1, η(K_{1,q})=2q, and η(D_{q,q})=5q for 2≤q≤6, showing that a linear bound is plausible while ruling out constants at or below 2. The authors formulate the unresolved issue as whether η(T)=O(n) for every tree.

References

Determine the growth of \eta on graphs of bounded degree, bounded degeneracy, or planar graphs.

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Problem 3 (Sparse classes), Section 10 (Open problems)

Is \eta(T)=O(n) for every tree T?

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Problem 1 (Trees), Section 10 (Open problems)

Is computing \eta NP-hard? Is \eta(G)\le N hard for N given in unary?

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Problem 5 (Complexity), Section 10 (Open problems)

We state this as a conjecture rather than a theorem.

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Conjecture 2, Section 8.1 (The double star family)