# On complete subgraphs of r-chromatic graphs

## Abstract

Let Gr(n) be an r-chromatic graph with n vertices in each colour class. Suppose G = G3(n), and δ(G), the minimal degree in G, is at least n + t(r ≥ 1). We prove that G contains at least t3 triangles but does not have to contain more than 4t3 of them. Furthermore, we give lower bounds for s such that G contains a complete 3-partite graph with s vertices in each class. Let f{hook}r(n) = max {δ(G): G = Gr(n), G does not contain a complete graph with r vertices}. We obtain various results on f{hook}r(n). In particular, we prove hat if cr = limn→∞ f{hook}r(n) n, then limr→∞(cr - (r - 2)) ≥ 1 2 and we conjecture that equally holds. We prove several other results and state a number of unsolved problems. © 1975.

## Publication Title

Discrete Mathematics

## Recommended Citation

Bollobás, B., Erdös, P., & Szemerédi, E.
(1975). On complete subgraphs of r-chromatic graphs.* Discrete Mathematics**, 13* (2), 97-107.
https://doi.org/10.1016/0012-365X(75)90011-4