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{\bf Jakub Przyby{\l}o}
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{\bf A Note on Neighbour-Distinguishing Regular Graphs Total-Weighting}
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We investigate the following modification of a problem posed by
Karo\'nski, \L uczak and Thomason [J. Combin. Theory, Ser. B 91
(2004) 151--157]. Let us assign positive integers to the edges and
vertices of a simple graph $G$. As a result we obtain a
vertex-colouring of $G$ by sums of weights assigned to the vertex
and its adjacent edges. Can we obtain a proper colouring using
only weights 1 and 2 for an arbitrary $G$?
We know that the answer is yes if $G$ is a 3-colourable, complete or
4-regular graph. Moreover, it is enough to use weights from $1$ to
$11$, as well as from $1$ to $\lfloor{\chi(G)\over2}\rfloor+1$,
for an arbitrary graph $G$. Here we show that weights from $1$ to
$7$ are enough for all regular graphs.
\bye