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The asymptotic behaviour of $sat(n,\mathcal{F})$

Asier Calbet, Andrea Freschi

Abstract

For a family $\mathcal{F}$ of graphs, $sat(n,\mathcal{F})$ is the minimum number of edges in a graph $G$ on $n$ vertices which does not contain any of the graphs in $\mathcal{F}$ but such that adding any new edge to $G$ creates a graph in $\mathcal{F}$. For singleton families $\mathcal{F}$, Tuza conjectured that $sat(n,\mathcal{F})/n$ converges and Truszczynski and Tuza discovered that either $sat(n,\mathcal{F})= \left(1-1/r\right)n+o(n)$ for some integer $r \geq 1$ or $ sat(n,\mathcal{F}) \geq n+o(n) $. This is often cited in the literature as the main progress towards proving Tuza's Conjecture. Unfortunately, the proof is flawed. We give a correct proof, which requires a novel construction. Moreover, for finite families $\mathcal{F}$, we completely determine the possible asymptotic behaviours of $sat(n,\mathcal{F})$ in the sparse regime $sat(n,\mathcal{F}) \leq n+o(n)$. Finally, we essentially determine which sequences of integers are of the form $\left(sat(n,\mathcal{F})\right)_{n \geq 0}$ for some (possibly infinite) family $\mathcal{F}$.

The asymptotic behaviour of $sat(n,\mathcal{F})$

Abstract

For a family of graphs, is the minimum number of edges in a graph on vertices which does not contain any of the graphs in but such that adding any new edge to creates a graph in . For singleton families , Tuza conjectured that converges and Truszczynski and Tuza discovered that either for some integer or . This is often cited in the literature as the main progress towards proving Tuza's Conjecture. Unfortunately, the proof is flawed. We give a correct proof, which requires a novel construction. Moreover, for finite families , we completely determine the possible asymptotic behaviours of in the sparse regime . Finally, we essentially determine which sequences of integers are of the form for some (possibly infinite) family .
Paper Structure (4 sections, 10 theorems, 22 equations, 17 figures)

This paper contains 4 sections, 10 theorems, 22 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Arbitrary families
  3. Quasi-finite families
  4. Singleton families

Key Result

Theorem 1

There exist sequences $(a_m)_{m \geq 0}$ and $(b_m)_{m \geq 0}$ of non-negative integers with $\exp\left[m \log m + O(m)\right] \geq a_m \geq b_m \geq \exp\left[m \log m + O(m \log \log m)\right]$ such that the following hold.

Figures (17)

  • Figure 1: The graph $H_k$.
  • Figure 2: The four types of graphs in $\mathcal{F}_I$.
  • Figure 3: The three types of graphs in $\mathcal{F}$.
  • Figure 4: The tree $\left(T' \cup kT\right)+v\mathbf{w}$.
  • Figure 5: All trees $T$ with $|T| \in \{1,2,3,4,5,7\}$.
  • ...and 12 more figures

Theorems & Definitions (29)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • ...and 19 more