# Distance two labeling for the strong product of two infinite paths

###
Yoomi Rho

Incheon National university

####
Byeong Kim

Gangneung-Wonju National University

####
Byung Song

Gangneung-Wonju National University

PDF

**Minisymposium:**
GRAPH PRODUCTS

**Content:**
An $L(h,k)$-labeling of a graph $G=(V,E)$ is a function $f$ on $V$
such that $|f(u)-f(v)|\ge h$ if ${\rm dist}(u,v)=1$ and
$|f(u)-f(v)|\ge k$ if ${\rm dist}(u,v)=2$. The span of $f$ is the
difference between the maximum and the minimum of $f$. The
$L(h,k)$-labeling number $\lambda_{h,k}(G)$ of $G$
is the minimum span over all $L(h,k)$-labelings of $G$.
Let $G$ be the strong product of two infinite paths.
Calamoneri found lower bounds and upper bounds of $\lambda_{h,k}(G)$ for all $h$ and $k$ when $h\ge k$.
Recently Kim et al. found a refined upper bound $3h+6k$ when $h\ge 4k$ and showed that it is the exact value when $h\ge 6k$.
In this paper, we obtain lower bounds and upper bounds when $h<k$.
Also we improve the result of Calamoneri when $k\le h\le 2k$.