K\"{u}hn and Osthus proved that, for every positive integer $\ell$, every $2^{16}\ell^2$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\geq \ell$ for every $v\in S$. And they asked whether the quadratic bound can be replaced by a linear bound. In this paper, we answer this question in the affirmative by proving that every $641\ell$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\ge \ell$ for every $v\in S$. The proof integrates the Moser-Tardos resampling algorithm.
Given a graph $G$ and a positive integer $k$, a packing $k$-domatic coloring of $G$ is a function $f: V(G) \to \{1,\ldots,t\}$ such that (1) for any color $j \in \{1,\ldots, t\}$ and any two distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=j$ we have $d_G(u,v)>j$, and (2) $V(G)$ admits a partition into $k$ sets $A_1,\ld...
We study a vertex-labeling analogue of the $1$-$2$-$3$ problem and its list version. For a labeling $\ell:V(G)\to\mathbb N$, let $S_\ell(v)=\sum_{w\in N(v)}\ell(w)$. The additive number $\eta(G)$ is the least $k$ for which there exists $\ell:V(G)\to[k]$ such that $S_\ell(u)\ne S_\ell(v)$ for every $uv\in E(G)$, while t...
For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Gerbner, Gy\H{o}ri, Methuku, and Vizer proved that $\text{ex}(n,C_{2\ell},C_{2k})=\Theta(n^\ell)$ for $k>\ell\ge2$. They determined the leading term for $\ell=2$, but for $k>\ell\ge3$ their general...
For a graph $G$, let $\text{cg}(G)$ be the maximum $k\in \mathbb N$ for which $G$ contains a $k$-vertex cograph as an induced subgraph. The Erd\H{o}s-Hajnal conjecture asserts that for every proper hereditary graph class $\mathcal{G}$, there exists $\gamma\in(0,1]$ such that every $n$-vertex graph $G\in\mathcal{G}$ sat...
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
Let $P_t$ denote the induced path on $t$ vertices. Let $\omega(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gy\'arf\'as (1987) proved that every $P_t$-free graph $G$ satisfies $\chi(G)\le(t-1)^{\omega(G)-1}$, and Gravier, Ho\`ang, and Maffray (2003) improved this to $\chi(G)\le (t-2)^{\omega(G)...
Sang-il Oum· 1 citation
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