Perfect Divisibility, Linear Divisibility and Chair-Free Graphs · arXivDesk
2608.14519Aug 14, 202618 pages. The chair-free result was obtained independently before we became aware of a recent preprint of Liu, Sun, Wang, Wu, and Zeng arXiv:2608.13519
Perfect Divisibility, Linear Divisibility and Chair-Free Graphs
A graph is perfectly divisible if every induced subgraph with at least one edge admits a partition into a perfect induced subgraph and an induced subgraph with smaller clique number. Every perfectly divisible graph G satisfies χ(H)≤(2ω(H)+1)
Nearby in the stack
for every induced subgraph
H
of
G
. We show that the converse fails: for every non-negative integer
t
, the graph
P(17)∨Kt
satisfies this bound for every induced subgraph but is not perfectly divisible, yielding an infinite family of counterexamples. Motivated by this distinction, we introduce
(k,ℓ)
-linear divisibility and prove that every
(k,ℓ)
-linearly divisible graph
G
satisfies
χ(G)≤k(2ω(G)+1)
. As an application of this framework, we give a direct structural decomposition showing that every chair-free graph is
(2,2)
-linearly divisible, where a chair is obtained from
K1,3
by subdividing one edge once. This chair-free result was obtained independently before we became aware of a recent preprint of Liu, Sun, Wang, Wu, and Zeng [arXiv:2608.13519], who prove the stronger statement that every chair-free graph is perfectly weight divisible and hence satisfies