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.
[Math] Clique problem for regular graphs
co.combinatoricsgraph theory
Best Answer
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