最大度为3的毛毛虫树的L(3,2,1)-标号  

L(3,2,1)-labeling of caterpillars with the maximum degree 3

作　　者：张小玲 ZHANG Xiaoling(Teachers College,Jimei University,Xiamen 361021,China)

机构地区：[1]集美大学师范学院,福建厦门361021

出　　处：《厦门大学学报(自然科学版)》2025年第2期356-360,366,共6页Journal of Xiamen University:Natural Science

基　　金：国家自然科学基金(12271210,11601265);集美大学科研启动基金(Q202201)。

摘　　要：[目的]对最大度为3的毛毛虫树的L(3,2,1)-标号问题进行研究.[方法]根据对毛毛虫树的最大度为3的点间距离进行分类,得到其可能的标号类型.利用这些可能的标号类型,通过拼接技术对最大度为3的毛毛虫树的L(3,2,1)标号数进行完全刻画.[结果]完全确定了最大度为3的毛毛虫树的L(3,2,1)标号数.[结论]本文的研究工作是先前文章《毛毛虫树的L(3,2,1)-标号问题》的一个延续.前文纠正了Clipperton在2008年发表的关于毛毛虫树的一个错误结果,并完全确定了最大度不小于4的毛毛虫树的L(3,2,1)标号数.本文则完全确定了最大度为3的毛毛虫树的L(3,2,1)标号数.这样对于毛毛虫树的L(3,2,1)-标号就得到了完全的刻画.[Objective]As a generalization of distance two labeling,multilevel distance labeling is motivated by the channel assignment problem.More recently,researchers began to investigate the L(3,2,1)-labeling problem.For example,Clipperton et al.proved that the L(3,2,1)-labeling number of a caterpillar with maximum degreeΔcan secure one of two values:2Δ+1 and 2Δ+2.Furthermore,we provide a complete characterization of the L(3,2,1)-labeling of caterpillars withΔ≥4.As a continuation of previous research,we determine the L(3,2,1)-labeling number of caterpillars with the maximum degree 3.[Methods]Let f be a 7-L(3,2,1)-labeling of a caterpillar T with the maximum degree 3.Based on the distance of two 3-vertex in T,we obtain their possible types,i.e.XX-type and XY-type,where X,Y∈{A,B,C}.Using these possible types,the L(3,2,1)-labeling number of caterpillars with the maximum degree 3 is obtained through concatenation.[Results]In this article,we obtain that the L(3,2,1)-labeling number of caterpillars with the maximum degree 3 depends on the type of T.The result is presented below.Let T be a caterpillar with the maximum degree 3 and a 1 a 2…a k-1 be the type of T.Then the L(3,2,1)-labeling number of T is 8 if and only if T contains one of following configurations given below.(C1)There is a i=2 for some i∈[1,k-1].(C2)There is a i=a j=1 such that a l∈{4,5,6,7,10}(if exists)for all l∈[i+1,j-1]and the number of consecutive 5 and the number of consecutive 7 in the set{a l|l∈[i+1,j-1]}are always even.(C3)There is a i=a j=3 such that a l∈{4,7,8}(if exists)for all l∈[i+1,j-1]and the number of consecutive 7 in the set{a l|l∈[i+1,j-1]}is always even.(C4)There is a i=1,a j=3 or a i=3,a j=1 such that a l∈{4,7}for all l∈[i+1,j-1]and the number of 7 in the set{a l|l∈[i+1,j-1]}is odd.[Conclusion]The result in the present article constitutes a continuation of the previous article entitled“The L(3,2,1)-labeling problem on caterpillars”.The previous article corrected an erroneous result about L(3,2,1)-labeling of

关 键 词：频率分配 L(3 2 1)-标号 毛毛虫树 

分 类 号：O157.5[理学—数学]