More on Rainbow Cliques in Edge-Colored Graphs

TL;DR

This paper establishes conditions on edge and color counts in large edge-colored graphs that guarantee many rainbow cliques of size k, and characterizes extremal graphs avoiding such cliques for specific k.

Contribution

It provides new bounds ensuring the abundance of rainbow cliques and characterizes extremal graphs without rainbow cliques for k=4,5, advancing understanding in edge-colored graph theory.

Findings

For large n, graphs with sufficiently high e(G)+c(G) contain Ω(n^k) rainbow K_k.

Characterization of extremal graphs without rainbow K_4 and K_5.

Addresses and completes previous research on rainbow cliques in edge-colored graphs.

Abstract

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any $\varepsilon>0$, if $e(G)+c(G) \geq (1+\frac{k-3}{k-2}+2\varepsilon) {n\choose 2}$ and $k\geq 3$, then for sufficiently large $n$, the number of rainbow cliques $K_k$ in $G$ is $\Omega(n^k)$. We also characterize the extremal graphs $G$ without a rainbow clique $K_k$, for $k=4,5$, when $e(G)+c(G)$ is maximum. Our results not only address existing questions but also complete the findings of Ehard and Mohr (Ehard and Mohr, Rainbow triangles and cliques in edge-colored graphs. {\it European Journal of Combinatorics, 84:103037,2020}).