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For every a bounded number of disjoint -restricted sets covers all but vertices of an -free graph
Statement
Fix a graph , a real , and . Then there is an integer such that every nonempty -free finite simple graph contains pairwise disjoint -restricted sets for some , whose union misses fewer than vertices.
Facts & Assumptions
Given: A graph , reals and .
Every induced subgraph of an -free graph is -free (Every class defined by forbidden induced subgraphs is hereditary, Every induced subgraph of an -free graph is -free, -free and -free graphs under the induced-subgraph convention).
Rödl's theorem provides a constant such that every nonempty -free graph contains an -restricted set of size at least times its order (Rödl: for every and every there is such that every nonempty -free graph has an -restricted vertex set of size at least ).
Sparsity is internal to induced subgraphs, so a set that is -restricted in an induced subgraph is also -restricted in the ambient graph (A set is -sparse exactly when the maximum degree of the graph it induces is at most times its size, -sparse, -dense and -restricted vertex sets).
Proof
Let be the constant from [L2]. Choose minimal with .
Starting from , repeatedly apply [L2] to the induced subgraph on the current remainder while that remainder has at least vertices. By [L1] each remainder is still -free, so this produces pairwise disjoint -restricted sets .
After each extraction, at least a fraction of the current remainder is removed. Hence after steps the remainder has size at most . In particular, after steps it has size strictly below by the choice of .
The process therefore stops after some number of extractions, and [L3] makes every extracted set -restricted in the original graph.
Depends on
- Rödl: for every $H$ and every $\epsilon\in(0,\tfrac12)$ there is $\delta>0$ such that every nonempty $H$-free graph has an $\epsilon$-restricted vertex set of size at least $\delta|V(G)|$
- A set is $c$-sparse exactly when the maximum degree of the graph it induces is at most $c$ times its size
- $c$-sparse, $c$-dense and $c$-restricted vertex sets
- Every class defined by forbidden induced subgraphs is hereditary
- Every induced subgraph of an $\mathcal F$-free graph is $\mathcal F$-free
- $H$-free and $\mathcal F$-free graphs under the induced-subgraph convention
- Subgraphs, induced subgraphs and spanning subgraphs
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- M. Chudnovsky, A. Scott, P. Seymour, and S. Spirkl, Strengthening Rödl's theorem, proof sketch of Theorem 1.3 (standard reference, not scraped)