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A -sparse set satisfies , and a -dense set satisfies
Statement
Let be a finite simple graph, let , and let be nonempty.
- If is -sparse, then .
- If is -dense, then .
Facts & Assumptions
Given: A finite simple graph , a real , and a nonempty set .
If is -sparse, then every vertex of has degree at most (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).
A graph of maximum degree at most has chromatic number at most (The greedy colouring bound for every nonnull finite graph, Proper vertex colourings and chromatic number).
Every finite graph satisfies (The bounds and , Cliques, independent sets, clique number and independence number).
The published definitions Cliques, independent sets, clique number and independence number and Cliques, stable sets, the clique number and stability number define the same invariants and under the same symbols.
In the complement graph, stable sets become cliques and sparse sets become dense sets by Complementation swaps cliques with stable sets, so and A set is -sparse in exactly when it is -dense in , so -restrictedness is complement-invariant.
Proof
In the sparse case, [L1] gives , so [L2] gives .
The two published definitions of and agree by [L4], so the complement statement can be read with the same symbols.
Applying [L3] to yields , hence .
If is -dense, then [L5] makes -sparse in , so step 2.1 applied there gives a stable set of size at least . Reading that set back in via [L5] gives a clique of the same size.
Depends on
- $c$-sparse, $c$-dense and $c$-restricted vertex sets
- A set is $c$-sparse exactly when the maximum degree of the graph it induces is at most $c$ times its size
- A set is $c$-sparse in $G$ exactly when it is $c$-dense in $\overline G$, so $c$-restrictedness is complement-invariant
- The greedy colouring bound $\chi(G)\leq\Delta(G)+1$ for every nonnull finite graph
- The bounds $\omega(G)\leq\chi(G)$ and $|V(G)|\leq\chi(G)\alpha(G)$
- Cliques, independent sets, clique number and independence number
- Proper vertex colourings and chromatic number
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Complementation swaps cliques with stable sets, so $\omega(\overline G)=\alpha(G)$
- Graph isomorphisms, automorphisms and graph complements
- Cliques, stable sets, the clique number $\omega(G)$ and stability number $\alpha(G)$
Used by
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