Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rainbow saturation for complete graphs

Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins

Abstract

We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph $Γ$ is called $H$-rainbow saturated if $Γ$ does not contain a rainbow copy of $H$ and adding an edge of any color to $Γ$ creates a rainbow copy of $H$. The rainbow saturation number $sat(n,{R}(H))$ is the minimum number of edges in an $n$-vertex $H$-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that $sat(n,{R}(K_r))=2(r-2)n+O(1)$ for fixed $r\geq 3$. Disproving this conjecture, we establish that for every $r\geq 3$, there exists a constant $α_r$ such that $$r + Ω\left(r^{1/3}\right) \le α_r \le r + r^{1/2} \qquad \text{and} \qquad sat(n,{R}(K_r)) = α_r n + O(1).$$ Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number, and asked whether this is equal to the rainbow saturation number of $K_r$, since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant $α_r$ resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored $K_r$-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are $K_r$-saturated with respect to the operation of deleting one edge and adding two edges.

Rainbow saturation for complete graphs

Abstract

We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph is called -rainbow saturated if does not contain a rainbow copy of and adding an edge of any color to creates a rainbow copy of . The rainbow saturation number is the minimum number of edges in an -vertex -rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that for fixed . Disproving this conjecture, we establish that for every , there exists a constant such that Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number, and asked whether this is equal to the rainbow saturation number of , since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored -rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are -saturated with respect to the operation of deleting one edge and adding two edges.
Paper Structure (11 sections, 30 theorems, 46 equations, 3 figures)

This paper contains 11 sections, 30 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Central lemma
  3. Rainbow Complete Graphs --- Proof of Theorem \ref{['thm1']}
  4. Reducing $\mathcal{R}(K_r)$-saturation to an equivalent problem
  5. Upper bound constructions for $f(k)$ and calculation of small values
  6. Lower bound of $f(k)$
  7. Non-stability for rainbow saturation
  8. $(H,1)$-saturation and -semisaturation, with applications to rainbow saturation problems
  9. $(K_r,1)$-semisaturation and $\mathcal{R}(K_r)$-semisaturation
  10. $(K_r,1)$-saturation and $(\mathcal{R}(K_n),\mathcal{R}(K_r))$-saturation
  11. Concluding remarks

Key Result

Theorem 1.1

For every $n\ge r-2$, Moreover, the unique graph witnessing the saturation number of $K_r$ is $K_{r-2}+\overline{K_{n-r+2}}$.

Figures (3)

  • Figure 1: Three edge graphs on five vertices
  • Figure 2: The graphs $\Lambda_3$ (left) and $\Lambda'_3$ (right). The dotted lines represent non-edges, and each gray edge has a unique color.
  • Figure 3: A $\mathcal{R}(K_3)$-saturated graph with exactly one missing (dotted) edge, denoted $\Lambda'_3$

Theorems & Definitions (53)

  • Theorem 1.1: Bollobas B65, Erdős--Hajnal--Moon EHM64, Zykov Z49
  • Conjecture 1.2: GLP20
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • Theorem 1.11
  • ...and 43 more