#### Vol. 8, No. 3, 2015

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On distance labelings of amalgamations and injective labelings of general graphs

### Nathaniel Karst, Jessica Oehrlein, Denise Sakai Troxell and Junjie Zhu

Vol. 8 (2015), No. 3, 535–540
##### 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 $G$.

##### Keywords
$L(2,1)$-labeling, distance two labeling, injective $L(2,1)$-labeling, amalgamation of graphs, channel assignment problem
##### Mathematical Subject Classification 2010
Primary: 68R10, 94C15
Secondary: 05C15, 05C78
##### Milestones 