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Ruszinkó, Miklós
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- PublicationProper edge colorings of planar graphs with rainbow C4 ${C}_{4}$‐s(2024)
; Sárközy, Gábor N.We call a proper edge coloring of a graph G $G$ a B-coloring if every 4-cycle of G $G$ is colored with four different colors. Let q B ( G ) ${q}_{B}(G)$ denote the smallest number of colors needed for a B-coloring of G $G$. Motivated by earlier papers on B-colorings, here we consider q B ( G ) ${q}_{B}(G)$ for planar and outerplanar graphs in terms of the maximum degree ? = ? ( G ) ${\rm{\Delta }}={\rm{\Delta }}(G)$. We prove that q B ( G ) ≤ 2 ? + 8 ${q}_{B}(G)\le 2{\rm{\Delta }}+8$ for planar graphs, q B ( G ) ≤ 2 ? ${q}_{B}(G)\le 2{\rm{\Delta }}$ for bipartite planar graphs, and q B ( G ) ≤ ? + 1 ${q}_{B}(G)\le {\rm{\Delta }}+1$ for outerplanar graphs with ? ≥ 4 ${\rm{\Delta }}\ge 4$. We conjecture that, for ? ${\rm{\Delta }}$ sufficiently large, q B ( G ) ≤ 2 ? ( G ) ${q}_{B}(G)\le 2{\rm{\Delta }}(G)$ for planar G $G$, and q B ( G ) ≤ ? ( G ) ${q}_{B}(G)\le {\rm{\Delta }}(G)$ for outerplanar G $G$.7