An extension of Nash-Williams and Tutte's Theorem

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-09-26 DOI:10.1002/jgt.23189

Xuqian Fang, Daqing Yang

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引用次数: 0

Abstract

The celebrated Nash-Williams and Tutte's Theorem states that a graph $G$ contains $k$ edge-disjoint spanning trees if and only if $ν_{f} (G) \geq k$ , where

In this paper, we prove that, for integers $k \geq 0$ , $d \geq 1$ , if $ν_{f} (G) > k + \frac{d - 1}{d}$ , then $G$ contains $k$ edge-disjoint spanning trees and another forest $F$ with $∣ E (F) ∣ > \frac{d - 1}{d} (∣ V (G) ∣ - 1)$ , and if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Moreover, the bound of $ν_{f} (G)$ is sharp.

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纳什-威廉姆斯和图特定理的推广

著名的 Nash-Williams 和 Tutte 定理指出，当且仅当 ν f ( G ) ≥ k ${{nu }_{f}(G)\ge k$ 时，图 G $G$ 包含 k $k$ 边互不相交的生成树，在本文中，我们证明，对于整数 k ≥ 0 $k\ge 0$ , d ≥ 1 $d\ge 1$ , 如果 ν f ( G ) > k + d - 1 d ${{nu }_{f}(G)\gt k+\frac{d-1}{d}$ ，则 G $G$ 包含 k $k$ 边互不相交的生成树和另一个森林 F $F$ ，其 ∣ E ( F ) ∣ > d - 1 d ( ∣ V ( G ) ∣ - 1 ) $| E(F)| \gt \frac{d-1}{d}(|V(G)|-1)$，如果 F $F$ 不是生成树，那么 F $F$ 有一个至少有 d $d$ 边的部分。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .