# Clique problem for regular graphs

I am looking for NP complete results for cliques in regular graphs. For example is the general problem of determining if a regular graph on n vertices has an n/2 clique NP-complete? (obviously the question is interesting only if the degree is at least n/2). You could also ask the same question for say regular graphs of degree 3n/4.

• There is lots of information (though not an answer to your exact question) at en.wikipedia.org/wiki/Clique_problem. Jun 24, 2014 at 4:06

Let $G = (V,E)$ be a graph and $c ≤ |V |$ a positive integer. The independent sets of $G$ are precisely the cliques of the complementary graph $\overline{G}$.

INDEPENDENT SET

INSTANCE: Graph $G=(V,E)$, positive integer $c$.

QUESTION: Does $G$ contain an independent set of size $c$ or more,

The independent set problem remains NP-complete when restricted to 3-regular planar graphs.

Reference:

COMPUTERS AND Intractability, A Guide to the Theory of NP-Completeness

Michael R. Garey / David S. Johnson

page 194-195

• This answers the question for cliques of specified size, but it doesn't answer it for cliques of size n/2. 3-regular graphs have an independent set of size $n/2$ iff they are bipartite. Jun 25, 2014 at 19:42
• The key is that the complement of a regular graph is regular. Jun 28, 2021 at 13:34