A New Type of Magical Coloring
Bing Yao, Zhang Zhong-fu, Ming Yao, Jingwen Li · Advances in Mathematics · 2008
Motivated from the edge-magic total labelling defined by Kotzig and Rosa (1970), we define a nearly k-magic coloring f of a graph G with p vertices and q edges that is a bi-jection from V(G)∪E(G) to {1,2,…p+q} such that |f(u)+f(v)-f(uv)|≤k whenever uv ∈ E(G). We have some results on a k-magic coloring f of G which is a nearly κ-magic coloring of G and keeps f(u)+f(v)=k+f(uv) for uv ∈ E(G). Furthermore, we study some graphs with a supper k-magic coloring f that is κ-magic coloring and f(u)≤p for all u∈V(G). We find some ways to construct some graphs having κ-magic coloring or supper κ-magic coloring, and show some properties about such graphs. At the end of this paper, we conjecture that any tree has a supper κ-magic coloring.