Distance three labelings of trees

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• 作者：Ji艡í ; Fiala^a ; ^{fiala@kam.mff.cuni.cz} ; Petr A. Golovach^b ; ^{petr.golovach@durham.ac.uk} ; Jan Kratochví ; l^a ; ^{honza@kam.mff.cuni.cz} ; Bernard Lidický ; ^a ; ^{bernard@kam.mff.cuni.cz} ; Danië ; l Paulusma^b ; ^{daniel.paulusma@durham.ac.uk}
• 关键词：Distance constrained graph labeling ; Linear distance ; Circular distance
• 刊名：Discrete Applied Mathematics
• 出版年：2012
• 期刊代码：46_0166218x
• 类别：cp
• 出版时间：April, 2012
• 卷：160
• 期：6
• 页码：764-779
• 文件大小：495 K

摘要

An -labeling of a graph assigns nonnegative integers to the vertices of in such a way that labels of adjacent vertices differ by at least two, while vertices that are at distance at most three are assigned different labels. The maximum label used is called the span of the labeling, and the aim is to minimize this value. We show that the minimum span of an -labeling of a tree can be bounded by a lower and an upper bound with difference one. Moreover, we show that deciding whether the minimum span attains the lower bound is an -complete problem. This answers a known open problem, which was recently posed by King, Ras, and Zhou as well. We extend some of our results to general graphs and/or to more general distance constraints on the labeling.