树的块图和线图的最优射标号
Optimal Radio Labellings of Block Graphs and Line Graphs of Trees
摘要：
图$G$的无线电标记是映射$f$：$V(G) \rightarrow \{0, 1, 2,...\}$，使得$|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$对每对顶点$u$和$v$成立，其中$diam(G)$是$G$的直径，$d(u,v)$是$G$中$u$和$v$之间的距离。由$rn(G)$表示的$G$的无线电数是最小的$t$，使得$G$允许用$t=\max\{|f(v)-f(u)|: v, u \in V(G)\}$进行无线电标记。块图是使得每个块（导出的最大2连通子图）是完全图的图。本文给出了块图的无线数的一个下界。达到该界的块图称为下界块图。我们证明了下界块图的三个充分必要条件。此外，我们给出了一个图是下界块图的三个充分条件。应用所建立的界和条件，我们证明了几类块图是下界块图，包括水平正则块图和块的扩展星。图$G(V,E)$的线图以$E(G)$为顶点集，其中两个顶点相邻，如果它们是$G$中的关联边。我们将结果推广到树，树的线图是块图。我们证明了，如果一棵树是一个下界块图，那么在一定条件下，它的线图也是一个下界块图，反之亦然。因此，我们证明了许多已知的下界树的线图，不包括路径，是下界块图。
英文摘要：
A radio labeling of a graph $G$ is a mapping $f$ : $V(G) \rightarrow \{0, 1, 2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of vertices $u$ and $v$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest $t$ such that $G$ admits a radio labeling with $t=\max\{|f(v)-f(u)|: v, u \in V(G)\}$. A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is established. The block graph which achieves this bound is called a lower bound block graph. We prove three necessary and sufficient conditions for lower bound block graphs. Moreover, we give three sufficient conditions for a graph to be a lower bound block graph. Applying the established bound and conditions, we show that several families of block graphs are lower bound block graphs, including the level-wise regular block graphs and the extended star of blocks. The line graph of a graph $G(V,E)$ has $E(G)$ as the vertex set, where two vertices are adjacent if they are incident edges in $G$. We extend our results to trees as trees and its line graphs are block graphs. We prove that if a tree is a lower bound block graph then, under certain conditions, its line graph is also a lower bound block graph, and vice versa. Consequently, we show that the line graphs of many known lower bound trees, excluding paths, are lower bound block graphs.
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