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Degree conditions for disjoint path covers in digraphs

Ansong Ma, Yuefang Sun

TL;DR

This paper obtains a minimum semi-degree sufficient condition for the one-to-many $k$-DDPC problem on a digraph with order $n$, and shows that the bound for the minimum semi-degree is sharp when $n+k$ is odd and is sharp up to an additive constant 1 otherwise.

Abstract

In this paper, we study degree conditions for three types of disjoint directed path cover problems: many-to-many $k$-DDPC, one-to-many $k$-DDPC and one-to-one $k$-DDPC, which are intimately connected to other famous topics in graph theory, such as Hamiltonicity and $k$-linkage, and have a strong background of applications. Firstly, we get two sharp minimum semi-degree sufficient conditions for the unpaired many-to-many $k$-DDPC problem and a sharp Ore-type degree condition for the paired many-to-many $2$-DDPC problem. Secondly, we obtain a minimum semi-degree sufficient condition for the one-to-many $k$-DDPC problem on a digraph with order $n$, and show that the bound for the minimum semi-degree is sharp when $n+k$ is even and is sharp up to an additive constant 1 otherwise. Finally, we give a minimum semi-degree sufficient condition for the one-to-one $k$-DDPC problem on a digraph with order $n$, and show that the bound for the minimum semi-degree is sharp when $n+k$ is odd and is sharp up to an additive constant 1 otherwise.

Degree conditions for disjoint path covers in digraphs

TL;DR

This paper obtains a minimum semi-degree sufficient condition for the one-to-many -DDPC problem on a digraph with order , and shows that the bound for the minimum semi-degree is sharp when is odd and is sharp up to an additive constant 1 otherwise.

Abstract

In this paper, we study degree conditions for three types of disjoint directed path cover problems: many-to-many -DDPC, one-to-many -DDPC and one-to-one -DDPC, which are intimately connected to other famous topics in graph theory, such as Hamiltonicity and -linkage, and have a strong background of applications. Firstly, we get two sharp minimum semi-degree sufficient conditions for the unpaired many-to-many -DDPC problem and a sharp Ore-type degree condition for the paired many-to-many -DDPC problem. Secondly, we obtain a minimum semi-degree sufficient condition for the one-to-many -DDPC problem on a digraph with order , and show that the bound for the minimum semi-degree is sharp when is even and is sharp up to an additive constant 1 otherwise. Finally, we give a minimum semi-degree sufficient condition for the one-to-one -DDPC problem on a digraph with order , and show that the bound for the minimum semi-degree is sharp when is odd and is sharp up to an additive constant 1 otherwise.
Paper Structure (9 sections, 15 theorems, 57 equations, 2 figures)

This paper contains 9 sections, 15 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation, terminology and notation
  3. Main results
  4. Many-to-many $k$-DDPC problem
  5. One-to-many $k$-DDPC problem
  6. One-to-one $k$-DDPC problem
  7. Many-to-many $k$-DDPC problem
  8. One-to-many $k$-DDPC problem
  9. One-to-one $k$-DDPC problem

Key Result

Theorem 1

Let $D$ be a digraph of order $n \geq 3k$, where $k$ is a positive integer. If $\delta^{0}(D) \geq \lceil (n + k)/2 \rceil$, then $D$ is unpaired many-to-many $k$-coverable. Moreover, the bound for $\delta^{0}(D)$ is sharp.

Figures (2)

  • Figure 1: The graph of Proposition \ref{['Pro:2t2']}.
  • Figure 2: The figure of the claim. The $s$-$t$ path is a Hamiltonian path. The thicker lines express two new $s$-$t$ paths: $P_{1} = sw^{+}Pt$ and $P_{2} = sPwt$.

Theorems & Definitions (27)

  • Definition 1: Paired/unpaired many-to-many $k$-coverable digraphs
  • Definition 2: One-to-many/one $k$-coverable digraphs
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 17 more