For a graph $F$, its extremal number ${\rm ex}(n,F)$ and extremal graphs denote the maximum number of edges in $F$-free graphs and the graphs attaining this maximum, respectively. Let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with ${\rm ex}(n,F)+q$ edges, and let $t_F(n,q)$ denote the cor...
Motivated by Mader's conjecture on connectivity keeping trees, we study the corresponding problem in triangle-free graphs. For positive integers $k,m$, we first prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\left \lceil \frac{m}{2} \right \rceil $ contains a path $P$ of order $m$ such tha...
Meng-Han Ma, Qing-Hai Liu, Li-Ping Zhang et al.· 0 citations
Motivated by Mader's conjecture on connectivity keeping trees, we study trees obtained from paths by adding one pendant vertex, as well as related problems in triangle-free graphs. For an integer $m$ and $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex a...
Meng-Han Ma, Qing-Hai Liu, Li-Ping Zhang et al.· 0 citations
For a graph $H$, the chromatic threshold $\delta_\chi(H)$ is the infimum of $c>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $cn$ is bounded by a constant depending only on $H$ and $c$. Allen, B\"ottcher, Griffiths, Kohayakawa, and Morris proved that if $\chi(H)=r\geq...
Jin-Ze Hu, Qing-Hai Liu, Li-Ping Zhang et al.· 1 citation
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