L(3,2,1)-Labeling problems on trapezoid graphs

Abstract

In graph theory, one of the most important problems is graph labeling. Labeling of graph has a very wide range of applications in [Formula: see text]-ray crystallography, missile guidance, coding theory, signal processing, radar, data base management, astronomy, communication network addressing, circuit design, etc. An extension of [Formula: see text]-labeling ([Formula: see text]-labeling) is [Formula: see text]-labeling ([Formula: see text]-labeling), which is now becoming a well-studied problem due to its application in real life situations. [Formula: see text]-labeling problem and the importance of [Formula: see text]-labeling problem motivates us to consider [Formula: see text]-labeling problem for trapezoid graph. The [Formula: see text]-labeling of a graph [Formula: see text] is an assignment [Formula: see text] from [Formula: see text] to the set [Formula: see text] such that if [Formula: see text] then [Formula: see text], if [Formula: see text] then [Formula: see text] and if [Formula: see text] then [Formula: see text], where [Formula: see text] is the distance between the nodes [Formula: see text] and [Formula: see text]. The [Formula: see text]-labeling number of a graph is denoted by [Formula: see text], it is the highest label used to label the graph [Formula: see text]. In this paper, for trapezoid graph [Formula: see text], it is proved that [Formula: see text], where [Formula: see text] is the degree of the graph [Formula: see text]. This upper bound is the first upper bound for [Formula: see text]-labeling problem on trapezoid graph. Also, to label any trapezoid graph, we have designed an algorithm which maintains the upper bound of the label.