On the Strong Path Partition Conjecture

IF 0.5 4区 数学 Q3 MATHEMATICS
J. de Wet, M. Frick, O. Oellermann, Jean E. Dunbar
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引用次数: 0

Abstract

Abstract The detour order of a graph G, denoted by τ (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that τ (〈A〉) ≤ a and τ (〈B〉) ≤ b, we say that (A, B) is an (a, b)-partition of G. If equality holds in both instances, we call (A, B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that τ (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a ≤ 8.
关于强路径划分猜想
图G的绕行阶,用τ(G)表示,是G中最长路径的阶。如果a和b是正整数,并且G的顶点集可以划分为两个子集a和b,使得τ(〈a〉)≤a和τ(〈b〉)≤b,我们说(a,b)是G的(a,b)-划分。路径分区猜想(PPC)断言,如果G是任何图,a,b是任何一对正整数,使得τ(G)=a+b,则G具有(a,b)-分区。强PPC断言,在相同的情况下,G有一个精确的(a,b)-分区。虽然在过去的三十年里,已经发展了大量支持PPC的工作,但文献中还没有出现关于强PPC的结果。本文证明了当a≤8时强PPC成立。
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
CiteScore
2.20
自引率
0.00%
发文量
22
审稿时长
53 weeks
期刊介绍: The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.
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