L(p,1)-labelling of graphs
Abstract
An $L(p,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq p$ if $\dist(x,y)=1$ and $|f(x)-f(y)|\geq p$ if $\dist(x,y)=2$, where $\dist(x,y)$ is the distance between the two vertices~$x$ and~$y$ in the graph. The \emph{span} of an $L(p,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\D^2+1$. We settle this conjecture for $\D$ 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 $\Delta\geq \Delta_p$ has an $L(p,1)$-labelling with span at most $\D^2+1$. This yields that, for each positive integer $p$, there is an integer $C_p$ such that every graph with maximum degree $\Delta$ has an $L(p,1)$-labelling with span at most $\Delta^2+C_p$.
Domains
Discrete Mathematics [cs.DM]Origin | Files produced by the author(s) |
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