Planar graphs with $\Delta\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

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Abstract

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list edge $\Delta$-colorable or list total $(\Delta+1)$-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta\geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta\geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k$ ($\forall 3 \leq k \leq 6$) are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.