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Bounds for Rainbow-uncommon Graphs

Blake Bates, Zhanar Berikkyzy, Nick Chiem, Gabriel Elvin, Risa Fines, Maja Lie, Hanna Mikulás, Isaac Reiter, Kevin Zhou

Abstract

We say a graph $H$ is $r$-rainbow-uncommon if the maximum number of rainbow copies of $H$ under an $r$-coloring of $E(K_n)$ is asymptotically (as $n \to \infty$) greater than what is expected from uniformly random $r$-colorings. Via explicit constructions, we show that for $H\in\{K_3,K_4, K_5\}$, $H$ is $r$-rainbow-uncommon for all $r\geq {|V(H)|\choose 2}$. We also construct colorings to show that for $t \geq 6$, $K_t$ is $r$-rainbow-uncommon for sufficiently large $r$.

Bounds for Rainbow-uncommon Graphs

Abstract

We say a graph is -rainbow-uncommon if the maximum number of rainbow copies of under an -coloring of is asymptotically (as ) greater than what is expected from uniformly random -colorings. Via explicit constructions, we show that for , is -rainbow-uncommon for all . We also construct colorings to show that for , is -rainbow-uncommon for sufficiently large .
Paper Structure (3 sections, 8 theorems, 19 equations, 2 figures)

This paper contains 3 sections, 8 theorems, 19 equations, 2 figures.

Table of Contents

  1. Background
  2. Main Result
  3. Conclusion

Key Result

Theorem 1

$K_t$ is ${t\choose{2}}$-rainbow-uncommon for all $t \geq 3$.

Figures (2)

  • Figure 1: Iterated blowup of a $K_4$. The "thick" edges indicate that all possible edges between the sets of vertices are given the specified color.
  • Figure 2: Parallel $r$-coloring defined in the proof of Proposition \ref{['prop-K3']}. In particular, there can be no non-rainbow triangles.

Theorems & Definitions (14)

  • Theorem 1: Erdos-HajnalDeSilva
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 4 more