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The weak saturation number of $\boldsymbol{K_{2, t}}$

Meysam Miralaei, Ali Mohammadian, Behruz Tayfeh-Rezaie

TL;DR

This paper studies the weak saturation number $\mathrm{wsat}(n,F)$ for bipartite graphs, with explicit focus on $F=K_{s,t}$ and, in particular, $F=K_{2,t}$. The authors derive a sharp, gcd- and parity-sensitive description of $\mathrm{wsat}(s+t, K_{s,t})$, showing it equals $\binom{s+t-1}{2}$ when $\gcd(s,t)=1$ and $\binom{s+t-1}{2}+1$ otherwise, using lower-bound arguments via the complement structure and matching upper-bound constructions. They then determine $\mathrm{wsat}(n, K_{2,t})$ for all $t\ge3$ and $n\ge t+2$ with a three-case formula based on the parity of $t$ and the size of $n$ relative to $2t-1$, supported by explicit saturated graphs, including the $\mathbb{G}_{n,t}$ family. The results complete the understanding of weak saturation for these bipartite patterns, rectify a prior claim in the literature, and connect to related themes in graph bootstrap percolation and random graph analogues.

Abstract

For two graphs $G$ and $F$, we say that $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ as a subgraph and one could join all the nonadjacent pairs of vertices of $G$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $\mathrm{wsat}(n, F)$ is the minimum number of edges of a weakly $F$-saturated graph on $n$ vertices. In this paper, we examine $\mathrm{wsat}(n, K_{s, t})$, where $K_{s, t}$ is the complete bipartite graph with parts of sizes $s$ and $ t $. We determine $\mathrm{wsat}(n, K_{2, t})$, correcting a previous report in the literature. It is also shown that $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}$ if $\gcd(s, t)=1$ and $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}+1$, otherwise.

The weak saturation number of $\boldsymbol{K_{2, t}}$

TL;DR

This paper studies the weak saturation number for bipartite graphs, with explicit focus on and, in particular, . The authors derive a sharp, gcd- and parity-sensitive description of , showing it equals when and otherwise, using lower-bound arguments via the complement structure and matching upper-bound constructions. They then determine for all and with a three-case formula based on the parity of and the size of relative to , supported by explicit saturated graphs, including the family. The results complete the understanding of weak saturation for these bipartite patterns, rectify a prior claim in the literature, and connect to related themes in graph bootstrap percolation and random graph analogues.

Abstract

For two graphs and , we say that is weakly -saturated if contains no copy of as a subgraph and one could join all the nonadjacent pairs of vertices of in some order so that a new copy of is created at each step. The weak saturation number is the minimum number of edges of a weakly -saturated graph on vertices. In this paper, we examine , where is the complete bipartite graph with parts of sizes and . We determine , correcting a previous report in the literature. It is also shown that if and , otherwise.
Paper Structure (3 sections, 12 theorems, 4 equations, 2 figures)

This paper contains 3 sections, 12 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Determination of $\boldsymbol{\mathrm{wsat}(s+t, K_{s,t})}$
  3. Determination of $\boldsymbol{\mathrm{wsat}(n, K_{2,t})}$

Key Result

Theorem 1.1

For every two integers $n, t$ with $t\geqslant3$ and $n\geqslant t+2$, the following statements hold.

Figures (2)

  • Figure 1: The graph $\mathbbmsl{G}_{n, t}$. We have not drawn the edges between the vertices in the gray elliptical disk.
  • Figure 2: The graph $\mathbbmsl{H}$. The set $X$ is a clique and the sets $Y_1, Y_2, Z$ are independent. Every vertex in $X$ is adjacent to every vertex in $Y_1\cup Y_2$ and every vertex in $Z$ is adjacent to every vertex in $Y_1$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 14 more