Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
Relating path coverings to vertex labellings with a condition at distance two
Discrete Mathematics
The $L(2,1)$-Labeling Problem on Graphs
SIAM Journal on Discrete Mathematics
A Theorem about the Channel Assignment Problem
SIAM Journal on Discrete Mathematics
Concentration for Independent Permutations
Combinatorics, Probability and Computing
Circular Distance Two Labeling and the $\lambda$-Number for Outerplanar Graphs
SIAM Journal on Discrete Mathematics
The Channel Assignment Problem with Variable Weights
SIAM Journal on Discrete Mathematics
Backbone colorings for graphs: Tree and path backbones
Journal of Graph Theory
Labeling planar graphs with a condition at distance two
European Journal of Combinatorics
Labelings of Graphs with Fixed and Variable Edge-Weights
SIAM Journal on Discrete Mathematics
Coloring the square of a planar graph
Journal of Graph Theory
Bounds for the Real Number Graph Labellings and Application to Labellings of the Triangular Lattice
SIAM Journal on Discrete Mathematics
Graph labellings with variable weights, a survey
Discrete Applied Mathematics
On Moore graphs with diameters 2 and 3
IBM Journal of Research and Development
The Computer Journal
Labeling outerplanar graphs with maximum degree three
Discrete Applied Mathematics
On the L(2,1)-labelings of amalgamations of graphs
Discrete Applied Mathematics
The L(p,q)-labelling of planar graphs without 4-cycles
Discrete Applied Mathematics
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An $L(p,1)$-labeling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geqslant p$ if dist$(x,y)=1$ and $|f(x)-f(y)|\geqslant 1$ if dist$(x,y)=2$, where dist$(x,y)$ is the distance between the two vertices $x$ and $y$ in the graph. The span of an $L(p,1)$-labeling $f$ is the difference between the largest and the smallest labels used by $f$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geqslant 2$ has an $L(2,1)$-labeling with span at most $\Delta^2$. We settle this conjecture for $\Delta$ sufficiently large. More generally, we show that for any positive integer $p$ there exists a constant $\Delta_p$ such that every graph with maximum degree $\Delta\geqslant \Delta_p$ has an $L(p,1)$-labeling with span at most $\Delta^2$. This yields that for each positive integer $p$, there is an integer $C_p$ such that every graph with maximum degree $\Delta$ has an $L(p,1)$-labeling with span at most $\Delta^2+C_p$.