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arXiv 1105.1906math.COcs.DM
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列出（$p$，1）的版本-标签总数

List version of ($p$,1)-total labellings

Yong Yu, Guanghui Wang, Guizhen Liu

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11 pages, 2 figures
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摘要：

图$G$的（$p$，1）-总数$λ_p^T(G)$是最小整数范围的宽度，该最小整数范围足以标记$G$的顶点和边，使得没有两个相邻顶点具有相同的标签，没有两个关联边具有相同的标签，并且顶点和其关联边的标签之间的差至少为$p$。在本文中，我们考虑列表版本。令$L(x)$为所有$x\in V(G)\cup E(G)$的可能颜色列表。将$C_{p,1}^T(G)$定义为最小整数$k$，使得对于所有$x\in V(G)\cup E(G)$具有$|L(x)|=k$的每个列表分配，$G$具有（$p$，1）-总标记$c$，使得所有$x\in V(G)\cup E(G)$的$c(x)\in L(x)$。我们称$C_{p,1}^T(G)$为（$p$，1）-全标记可选择性，$G$为列表$L$-（$p$，1）-全标记可选择性。本文提出了一个关于$C_{p,1}^T$的上界的猜想.此外，我们在第2节中研究了路径和树的这个参数。在第三节中我们还证明了具有$p\geq2, n\geq3$的星$K_{1,n}$的$C_{p,1}^T(K_{1,n})\leq n+2p-1$和具有$Δ\geq p+3$的外平面图的$C_{p,1}^T(G)\leq Δ+2p-1$.

英文摘要：

The ($p$,1)-total number $λ_p^T(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and the edges of $G$ such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least $p$. In this paper we consider the list version. Let $L(x)$ be a list of possible colors for all $x\in V(G)\cup E(G)$. Define $C_{p,1}^T(G)$ to be the smallest integer $k$ such that for every list assignment with $|L(x)|=k$ for all $x\in V(G)\cup E(G)$, $G$ has a ($p$,1)-total labelling $c$ such that $c(x)\in L(x)$ for all $x\in V(G)\cup E(G)$. We call $C_{p,1}^T(G)$ the ($p$,1)-total labelling choosability and $G$ is list $L$-($p$,1)-total labelable. In this paper, we present a conjecture on the upper bound of $C_{p,1}^T$. Furthermore, we study this parameter for paths and trees in Section 2. We also prove that $C_{p,1}^T(K_{1,n})\leq n+2p-1$ for star $K_{1,n}$ with $p\geq2, n\geq3$ in Section 3 and $C_{p,1}^T(G)\leq Δ+2p-1$ for outerplanar graph with $Δ\geq p+3$ in Section 4.

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