The $H$-linkage problems in sparse robustly expanding digraphs

TL;DR

This paper proves new degree sequence conditions that guarantee the existence of specific $H$-subdivisions and tilings in large digraphs, strengthening asymptotic versions of the Nash-Williams conjecture.

Contribution

It establishes sufficient degree conditions for $( ext{N}H)$-linkage and perfect $H$-subdivision tilings in large digraphs, extending prior asymptotic results.

Findings

Proves $( ext{N}H)$-linkage under degree sequence conditions.

Ensures existence of perfect $H$-subdivision tilings with minimum subdivision size.

Strengthens asymptotic Nash-Williams conjecture results.

Abstract

The Nash-Williams conjecture establishes degree sequence conditions ensuring Hamilton cycles in digraphs. An asymptotic version of this conjecture for large digraphs was independently derived by several researchers. We strengthen these results by proving the following results under the same asymptotic degree sequence conditions. For any digraph $H$, a digraph $D$ is $(\mathcal{N}H)$-linked if there exists an integer $l_0$ such that for any vertex set $U$ of cardinality $|V(H)|$ and every integer set $\mathcal{N}=\{l_i\}_{i=1}^{|A(H)|}$ with $l_i\geq l_0$, $D$ contains an $H$-subdivision with $U$ as branch-vertex set and the values in $\mathcal{N}$ specifying the lengths of the subdivided paths. Let $D$ be a sufficiently large digraph of order $n$ with the out-degree sequence $d_1^+\leq\cdots\leq d_n^+$ and the in-degree sequence $d_1^-\leq\cdots\leq d_n^-$. We prove that if for every $\gamma\in(0, 1)$ and every integer $0\leq i<n/2$, the following conditions hold: (i) $d_i^+\geq i+\gamma n$ or $d_{n-i-\gamma n}^-\geq n-i$, and (ii) $d_i^-\geq i+\gamma n$ or $d_{n-i-\gamma n}^+\geq n-i$, then $D$ is $(\mathcal{N}H)$-linked, and also admits a perfect $H$-subdivision tiling with subdivision orders $\{n_1, \ldots, n_k\}$, where each $n_i\geq C_0$ for some integer $C_0$.