论文标题
奇数树的设定标签
Set-Sequential Labelings of Odd Trees
论文作者
论文摘要
如果$ v（t）\ cup e（t）$中的元素可以用独特的非零$（n+1）$ - 尺寸$ 01 $ 01 $ - vectors标记为$ 2^n $顶点的树$ t $，则称为Set-Sequerential-sequerention-sequinential-cup e（t）$ - cup e（t）$。已经猜想，只有奇数度的$ 2^n $顶点上的所有树都是设定的（“奇特的猜想”），在本文中，我们提出了朝着猜想的进步。我们表明，某些类型的毛毛虫（对顶点的程度受到限制，但对直径没有限制）是设定的。此外，我们从较小的序列两部分图（不一定是奇数树）中介绍了一些新的设置序列图的构造。我们还猜想了$ \ mathbb {f} _2^n $的元素的配对；在此过程中，我们对Balister等人的2011年论文中的$ \ mathbb {f} _2^n $分区的定理证明进行了实质性的澄清。最后，我们在Balister等人的定理的两部分图上提出了结果。
A tree $T$ on $2^n$ vertices is called set-sequential if the elements in $V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional $01$-vectors such that the vector labeling each edge is the component-wise sum modulo $2$ of the labels of the endpoints. It has been conjectured that all trees on $2^n$ vertices with only odd degree are set-sequential (the "Odd Tree Conjecture"), and in this paper, we present progress toward that conjecture. We show that certain kinds of caterpillars (with restrictions on the degrees of the vertices, but no restrictions on the diameter) are set-sequential. Additionally, we introduce some constructions of new set-sequential graphs from smaller set-sequential bipartite graphs (not necessarily odd trees). We also make a conjecture about pairings of the elements of $\mathbb{F}_2^n$ in a particular way; in the process, we provide a substantial clarification of a proof of a theorem that partitions $\mathbb{F}_2^n$ from a 2011 paper by Balister et al. Finally, we put forward a result on bipartite graphs that is a modification of a theorem in Balister et al.
