On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

Abstract

Given an integer $r\geqslant 1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G \nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots, {H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory 11(1987), 191–196], we study the minimum number of edges over all $(K_t, \mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily $K_t$-saturated graph to prove that for all $t\geqslant4 $ and $k\geqslant\max\{6, t\}$, there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if $G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$ e(G)\geqslant \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2} \right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{4,5\}$ and $k\geqslant6$. The method we develop in this paper may shed some light on attacking Hanson and Toft's conjecture.

On the size of $(K_t, K_{1,k})$-co-critical graphs

“…, H k )-saturated graphs [4,8,13,14]. We refer the reader to a recent paper by Zhang and the second author [14] for further background on (H 1 , . .…”

“…Very recently, Zhang and the second author [14] obtained a lower bound for the size of (K t , T k )co-critical graphs for all t ≥ 4 and k ≥ max{6, t}. In addition, this bound is asymptotically best possible when t ∈ {4, 5} and all k ≥ 6 and n large.…”

On the size of special class 1 graphs and $(P_3; k)$-co-critical graphs

“…, H k )-saturated graphs [2,3,8,9]. We refer the reader to a recent paper by the last two authors [9] for further background on (H 1 , . .…”

“…Theorem 1.3 (Song and Zhang [9]) Let t, k ∈ N with t ≥ 4 and k ≥ max{6, t}. There exists a constant ℓ(t, k) such that, for all n ∈ N with n…”

“…The methods developed in [8,9] may shed some light on attacking Conjecture 1.1. Inspired by Conjecture 1.1, Ferrara, Kim and Yeager [3] proposed the following problem.…”