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On 0-Rotatable Graceful Caterpillars
Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2020-09-03 , DOI: 10.1007/s00373-020-02226-0
Atílio G. Luiz , C. N. Campos , R. Bruce Richter

An injection \(f :V(T) \rightarrow \{0,\ldots ,|E(T)|\}\) of a tree T is a graceful labelling if \(\{|f(u)-f(v)| :uv \in E(T)\}=\{1,\ldots ,|E(T)|\}\). Tree T is 0-rotatable if, for any \(v \in V(T)\), there exists a graceful labelling f of T such that \(f(v)=0\). In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star \(K_{1,s-1}\) to a leaf of a path \(P_n\) with \(n \ge 3\) and \(s \ge \lceil \frac{n}{2} \rceil \); caterpillars with diameter five or six; and caterpillars T with \(\mathrm {diam}(T) \ge 7\) such that, for every non-leaf vertex \(v \in V(T)\), the number of leaves adjacent to v is even and is at least \(2+2((\mathrm {diam}(T)-1)\bmod {2})\). These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.



中文翻译:

在0旋转的优美毛毛虫上

如果\(\ {| f(u)-f(是对树T的注入\(f:V(T)\ rightarrow \ {0,\ ldots,| E(T)| \} \)v)|:uv \ in E(T)\} = \ {1,\ ldots,| E(T)| \} \)。如果对于任何\(v(in V(T)\))存在一个优美的T标记f使得\(f(v)= 0 \),则树T是0旋转的。在这项工作中,以下毛毛虫科被证明是可0旋转的:具有完美匹配的毛毛虫;通过将星形\(K_ {1,s-1} \)的一片叶子链接到路径\(P_n \)的叶子与\(n \ ge 3 \)\(s \ ge \ lceil \ frac {n} {2} \ rceil \); 直径为五或六的毛毛虫;和带有\(\ mathrm {diam}(T)\ ge 7 \)的毛毛虫T,使得对于每个非叶子顶点\(v \ in V(T)\),与v相邻的叶子数是偶数,并且至少为\(2 + 2((\ mathrm {diam}(T)-1)\ bmod {2})\)。这些结果使人猜想所有直径至少为5的毛毛虫都可以0旋转。

更新日期:2020-09-03
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