Document Type

Article

Publication Date

1-1-2025

Journal / Book Title

Electronic Journal of Combinatorics

Abstract

Nordhaus and Gaddum proved sharp upper and lower bounds on the sum and product of the chromatic number of a graph and its complement. Over the years, similar inequalities have been shown for a plenitude of different graph invariants. In this paper, we consider such inequalities for the number of cliques (complete subgraphs) in a graph G, denoted k(G). We note that some such inequalities have been well-studied, e.g., lower bounds on k(G) + k(G) = k(G) + i(G), where i(G) is the number of independent subsets of G, has been come to be known as the study of Ramsey multiplicity. We give a history of such problems. One could consider fixed sized versions of these problems as well. We also investigate multicolor versions of these problems, meaning we r-color the edges of Kn yielding graphs G1, G2, …, Gr and give bounds on ∑ k(Gi) and ∏ k(Gi).

DOI

10.37236/13158

Rights

© The authors. Released under the CC BY-ND license (International 4.0)

Published Citation

Bal, D., Cutler, J., & Pebody, L. (2025). On the Number of Monochromatic Cliques in a Graph. The Electronic Journal of Combinatorics, 32(3), #P3.16. https://doi.org/10.37236/13158

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