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$$P_{3}$$ P 3 -Factors in the Square of a Tree
Graphs and Combinatorics ( IF 0.7 ) Pub Date : 2020-08-04 , DOI: 10.1007/s00373-020-02184-7
Guowei Dai , Zhiquan Hu

A spanning subgraph H of a graph G is a \(P_{3}\)-factor ofG if every component of H is a path of order three. The square of a graph G, denoted by \(G^{2}\), is the graph with vertex set V(G) such that two vertices are adjacent in \(G^{2}\) if and only if their distance in G is at most 2. A graph G is subcubic if it has maximum degree at most three. In this paper, we give a sharp necessary condition for the existence of \(P_{3}\)-factors in the square of a tree, which improves a result of Li and Zhang (Graphs Combin 24:107–111, 2008). In addition, we will also present a sufficient condition for the existence of \(P_{3}\)-factors in the square of a tree, which has the following interesting application to subcubic trees: if T is a subcubic tree of order 3n such that \(|L(T-L(T))|\le 7\), then \(T^{2}\) has a \(P_{3}\)-factor, where L(T) denotes the set of leaves in T. Examples show that the upper bound 7 on \(|L(T-L(T))|\) is sharp.



中文翻译:

$$ P_ {3} $$ P 3-树的正方形中的因子

甲生成子图ħ的曲线图的G ^是一个\(P_ {3} \) -的因子g ^如果的每个组件ħ是三阶的路径。图G的平方(用\(G ^ {2} \)表示是顶点集为VG)的图,使得当且仅当两个顶点在\(G ^ {2} \)中相邻。G中的距离最大为2。如果图G的最大程度最大为3,则它是三次的。在本文中,我们为\(P_ {3} \)的存在给出了清晰的必要条件影响树的平方,从而改善了Li和Zhang的效果(Graphs Combin 24:107-111,2008)。此外,我们还将为树的平方中存在\(P_ {3} \)因子提供充分的条件,这对于次立方树具有以下有趣的应用:如果T是3阶次立方树n使得\(| L(TL(T))| \ le 7 \),则\(T ^ {2} \)具有\(P_ {3} \)因子,其中LT)表示T中的叶子集。示例显示\(| L(TL(T))| \)的上限7是尖锐的。

更新日期:2020-08-04
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