Sufficient conditions for a planar graph to be list edge \(\Delta \)-colorable and list totally \((\Delta +1)\)-colorable (Q1939571)

The list chromatic index of a graph \(G\), \(\chi'_l(G)\), is the smallest integer \(l\) such that given any family of sets (lists) of colors \(L(x)\) assigned to the edges of \(G\) where \(|L(x)| \leq l\) for \(x\in E(G)\), it is possible to choose a color from every list resulting in proper edge-coloring of \(G\). Similarly, the list total chromatic number of \(G\), \(\chi''_l(G)\), is the smallest integer \(l\) such that given any family of sets (lists) of colors \(L(x)\) assigned to the edges and vertices of \(G\) where \(|L(x)| \leq l\) for \(x\in V(G) \cup E(G)\), it is possible to choose a color from every list resulting in such a coloring that each pair of adjacent or incident elements of \(V(G)\cup E(G)\) obtains distinct colors. The authors prove that \(\chi'_l = \Delta(G)\) and \(\chi''_l(G) = \Delta(G)+1\) for the planar graphs with the maximum degree \(\Delta(G) \geq 7\) (\(\Delta(G) \geq 8\), respectively) and not containing adjacent cycles of length \(4\) (\(3\) and \(5\), respectively).They also prove that \(\chi'_l(G) \leq \Delta(G)+1\) and \(\chi''_l(G) \leq \Delta(G)+2\) if \(\Delta(G) \geq 6\) and there are no adjacent cycles of length \(3\) and \(5\) in \(G\).