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Bipartitions with prescribed order of highly connected digraphs

Yuzhen Qi, Jin Yan, Jia Zhou

Abstract

A digraph is strongly connected if it has a directed path from $x$ to $y$ for every ordered pair of distinct vertices $x, y$ and it is strongly $k$-connected if it has at least $k+1$ vertices and remains strongly connected when we delete any set of at most $k-1$ vertices. For a digraph $D$, we use $δ(D)$ to denote $\mathop{\text{min}}\limits_{v\in V (D)} {|N_D^+(v)\cup N_D^-(v)|}$. In this paper, we show the following result. Let $k, l, n, n_1, n_2 \in \mathbb{N}$ with $n_1+n_2\leq n$ and $n_1,n_2\geq n/20$. Suppose that $D$ is a strongly $10^7k(k+l)^2\log(2kl$)-connected digraph of order $n$ with $δ(D)\geq n-l$. Then there exist two disjoint subsets $V_1, V_2\in V(D)$ with $|V_1| = n_1$ and $|V_2| = n_2$ such that each of $D[V_1]$, $D[V_2]$, and $D[V_1, V_2]$ is strongly $k$-connected. In particular, $V_1$ and $V_2$ form a partition of $V(D)$ when $n_1+n_2=n$. This result improves the earlier result of Kim, Kühn, and Osthus [SIAM J. Discrete Math. 30 (2016) 895--911].

Bipartitions with prescribed order of highly connected digraphs

Abstract

A digraph is strongly connected if it has a directed path from to for every ordered pair of distinct vertices and it is strongly -connected if it has at least vertices and remains strongly connected when we delete any set of at most vertices. For a digraph , we use to denote . In this paper, we show the following result. Let with and . Suppose that is a strongly )-connected digraph of order with . Then there exist two disjoint subsets with and such that each of , , and is strongly -connected. In particular, and form a partition of when . This result improves the earlier result of Kim, Kühn, and Osthus [SIAM J. Discrete Math. 30 (2016) 895--911].
Paper Structure (11 sections, 10 theorems, 29 equations, 5 figures)

This paper contains 11 sections, 10 theorems, 29 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and tools
  3. Basic terminology
  4. Some lemmas
  5. Proof of Theorem \ref{['theorem1']}
  6. Construction of almost dominating sets
  7. Construction of the correct $i$-path
  8. Coloring of the vertices in $E$
  9. Construction of sets $V_1$ and $V_2$
  10. Concluding remark
  11. Acknowledgements

Key Result

Theorem 1.1

Kang(2020) Let $k, t,l, m, n, q, a_1,\ldots, a_t \in \mathbb{N}$ with $t, m \geq 2$, $\sum_{i\in [t]} a_i \leq n$ and $a_i\geq n/(10tm)$ for each $i \in [t]$. Suppose that D is a strongly $10^8qk^2l(k+l)^2tm^2\log(m)$-connected digraph of order n with $\delta(D)\geq n-l$, and $Q_1,\ldots, Q_t \subse

Figures (5)

  • Figure 1: Color all vertices in $D_0$ in the case when $k = 1$.
  • Figure 2: The structure of $D[W_I\cup W_{II}]$ at Step 1 in the proof of Claim \ref{['claim1']}.
  • Figure 3: The structure of $D[W_I]$ at Step 2 in the proof of Claim \ref{['claim1']}.
  • Figure 4: The structure of $P_{i,j}$ with $(i,j)\in L\times L_0$.
  • Figure 5: Color patterns of the paths int($P_i$) with $i \in L\cap [6k]$ and $l=1$. The black arrows indicate $P_i$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Proof 1
  • Lemma 2.1
  • Proof 2
  • Lemma 2.2
  • Proposition 2.1
  • Theorem 2.1
  • ...and 29 more