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Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles
The Total Coloring Conjecture (TCC) is a challenging unsolved problem posed by Behzad and Vizing independently, which states that every simple graph $G$ admits a ($Δ(G)$ +2)-total-coloring, where $Δ(G)$ denotes the maximum degree of $G$. This conjecture has been confirmed for graphs with $Δ(G)\leq 5
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