For $k\ge2$, a graph is $C_{2k}$-connected if every two distinct edges lie together on an even cycle of length at most $2k$ in that graph. Let $f_k(n,m)$ be the largest number of edges in a $C_{2k}$-connected subgraph that every $n$-vertex graph with $m$ edges must contain. In this paper, we study two cycle-connected p...
An $r$-uniform hypergraph is linear if any two edges share at most one vertex. For a family $\mathcal F$ of $r$-graphs, let $\text{ex}^{\text{lin}}_r(n,\mathcal F)$ and $\text{spex}^{\text{lin}}_r(n,\mathcal F)$ denote, respectively, the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal...
We prove a sparse version of the four-colour theorem of Brandt and Thomass\'{e}, answering a question of Allen, B\"ottcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn...
Guo-Rong Gao, Jia-Lin He· 0 citations
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