arXiv subjects
Publications and source records attributed to Monika Pilsniak.
We consider edge colorings of a graph in such a way that each two different triangles have distinct colorings. It is an extension of the well-known idea of distinguishing all maximal stars in a graph. It was introduced in literature in 1985 and studied by many authors in various variants, but always for stars. We estimate new invariants regarding triangles for proper and general colorings.
With any (not necessarily proper) edge $k$-colouring $γ:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $σ\_γ$ given by $σ\_γ(v)=\sum\_{e\ni v}γ(e)$.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 $G$ admitsa 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 andtotal colourings, that is colourings $γ$ for whichthe number of elements in any two colour classes of $γ$ differ by at most one.We determine the equitable neighbour-sum-distinguishing indexof complete graphs, complete bipartite graphs and forests,and the equitable neighbour-sum-distinguishing total chromatic numberof complete graphs and bipartite graphs.