On distance labelings of amalgamations and injective labelings of general graphs

Abstract

An $L\left(2,1\right)$-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 $L\left(2,1\right)$-labelings of $G$ is denoted $\lambda \left(G\right)$. An $L^{\prime }\left(2,1\right)$-labeling of $G$ and the number ${\lambda }^{\prime }\left(G\right)$ are defined analogously, with the additional restriction that the labelings must be injective. We determine $\lambda \left(H\right)$ when $H$ is a join-page amalgamation of graphs, which is defined as follows: given $p\ge 2$, $H$ is obtained from the pairwise disjoint union of graphs $H_0,H_1,\dots ,H_p$ by adding all the edges between a vertex in $H_0$ and a vertex in $H_i$ for $i=1,2,\dots ,p$. Motivated by these join-page amalgamations and the partial relationships between $\lambda \left(G\right)$ and ${\lambda }^{\prime }\left(G\right)$ for general graphs $G$ provided by Chang and Kuo, we go on to show that ${\lambda }^{\prime }\left(G\right)=max\left\{n_G-1,\lambda \left(G\right)\right\}$, where $n_G$ is the number of vertices in $G$.