Proper edge colorings of planar graphs with rainbow C4-s
Abstract
We call a proper edge coloring of a graph G a B-coloring if every 4-cycle of G is colored with four different colors. Let qB(G) denote the smallest number of colors needed for a B-coloring of G. Motivated by earlier papers on B-colorings, here we consider qB(G) for planar and outerplanar graphs in terms of the maximum degree = (G). We prove that qB(G) 2+8 for planar graphs, qB(G) 2 for bipartite planar graphs and qB(G) +1 for outerplanar graphs with 4. We conjecture that, for sufficiently large, qB(G) 2(G) for planar G and qB(G) (G) for outerplanar G.
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