Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
Labeling Chordal Graphs: Distance Two Condition
SIAM Journal on Discrete Mathematics
On the $\lambda$-Number of $Q_n$ and Related Graphs
SIAM Journal on Discrete Mathematics
A Theorem about the Channel Assignment Problem
SIAM Journal on Discrete Mathematics
Labeling Planar Graphs with Conditions on Girth and Distance Two
SIAM Journal on Discrete Mathematics
A bound on the chromatic number of the square of a planar graph
Journal of Combinatorial Theory Series B
Coloring the square of a planar graph
Journal of Graph Theory
Labelling planar graphs without 4-cycles with a condition on distance two
Discrete Applied Mathematics
List 2-distance (Δ+2)-coloring of planar graphs with girth six
European Journal of Combinatorics
Graph labellings with variable weights, a survey
Discrete Applied Mathematics
The L(p,q)-labelling of planar graphs without 4-cycles
Discrete Applied Mathematics
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Wegner conjectured that for each planar graph $G$ with maximum degree $\Delta$ at least 4, $\chi(G^2)\leq\Delta+5$ if $4\leq\Delta\leq7$, and $\chi(G^2)\leq\lfloor \frac{3\Delta}{2}\rfloor +1$ if $\Delta\geq8$. Let $G$ be a planar graph without 4- and 5-cycles. In this paper, we discuss the $L(p,q)$-labeling of $G$ and show that $\lambda_{p,q}(G)\leq(2q-1)\Delta+6p+6q-6$ and $\lambda_{p,q}(G)\leq\max\{(2q-1)\Delta+6p+2q-4,9(2q-1)+8p-4,6(2q-1)+10p-5\},$ where $p$ and $q$ are positive integers with $p\geq q$. As a corollary, $\chi(G^2)\leq\Delta+7$ if $\Delta\leq7$, $\chi(G^2)\leq14$ if $\Delta=8$, and $\chi(G^2)\leq\Delta+5$ if $\Delta\geq9$.