L(2,1)-Edge-Labelings of the Edge-Path-Replacement of a Graph

Abstract

Two edges [Formula: see text] and [Formula: see text] in a graph [Formula: see text] are said to be adjacent if they have a vertex in common, and distance two apart if they are nonadjacent but both are adjacent to a common edge. An [Formula: see text]-edge-labeling of a graph [Formula: see text] is an assignment of nonnegative integers, called labels, to the edges of [Formula: see text] such that the difference between labels of any two adjacent edges is at least [Formula: see text], and the labels of any two edges that are distance two apart are different. The span of an [Formula: see text]-edge-labeling of a graph [Formula: see text] is the difference between the maximum and minimum labels. The minimum span over all [Formula: see text]-edge-labelings of a graph [Formula: see text] is called the [Formula: see text]-edge-labeling number of [Formula: see text], denoted by [Formula: see text]. For [Formula: see text], the edge-path-replacement of a graph [Formula: see text], denoted by [Formula: see text], is a graph obtained by replacing each edge of [Formula: see text] with a path [Formula: see text] on [Formula: see text] vertices. This paper investigates the [Formula: see text]-edge-labeling number of the edge-path-replacement [Formula: see text] of a graph [Formula: see text] for [Formula: see text]. We get the following main results: [Formula: see text] Let [Formula: see text] be a graph with maximum degree [Formula: see text] and [Formula: see text] be an integer not less than [Formula: see text], then [Formula: see text] if [Formula: see text] is odd, and otherwise [Formula: see text]. [Formula: see text] Let [Formula: see text] be a graph with maximum degree [Formula: see text]. Then [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd.