On distance labelings of amalgamations and injective labelings of general graphs

Abstract

An L.2; 1/-labeling of a graph G is a function assigning a nonnegative integer to each vertex such that adjacent vertices are labeled with integers differing by at least 2 and vertices at distance two are labeled with integers differing by at least 1. The minimum span across all L.2; 1/-labelings of G is denoted . G/. An L 0 .2; 1/-labeling of G and the number 0 .G/ are defined analogously, with the additional restriction that the labelings must be injective. We determine . H/ when H is a join-page amalgamation of graphs, which is defined as follows: given p 2, H is obtained from the pairwise disjoint union of graphs H0; H1;:::; Hp by adding all the edges between a vertex in H0 and a vertex in Hi for iD 1; 2;:::; p. Motivated by these join-page amalgamations and the partial relationships between. G/ and 0 .G/ for general graphs G provided by Chang and Kuo, we go on to show that 0 .G/D maxfnG 1;. G/g, where nG is the number of vertices in G.