Equitable neighbour-sum-distinguishing edge and total colourings
Résumé
With any (not necessarily proper) edge $k$-colouring $\gamma:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,
one can associate a vertex colouring $\sigma_{\gamma}$ given by $\sigma_{\gamma}(v)=\sum_{e\ni v}\gamma(e)$.
A neighbour-sum-distinguishing edge $k$-colouring is an edge colouring whose associated vertex colouring is proper.
The neighbour-sum-distinguishing index of a graph $G$ is then the smallest $k$ for which $G$ admits
a neighbour-sum-distinguishing edge $k$-colouring.
These notions naturally extends to total colourings of graphs that assign colours to both vertices and edges.
We study in this paper equitable neighbour-sum-distinguishing edge colourings and
total colourings, that is colourings $\gamma$ for which
the number of elements in any two colour classes of $\gamma$ differ by at most one.
We determine the equitable neighbour-sum-distinguishing index
of complete graphs, complete bipartite graphs and forests,
and the equitable neighbour-sum-distinguishing total chromatic number
of complete graphs and bipartite graphs.
Origine : Fichiers produits par l'(les) auteur(s)
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