List edge and list total colorings of planar graphs without 6-cycles with chord (Q2880475)

The author proves the following two results related to list edge and list total colorings of planar graphs without 6-cycles with chord.NEWLINENEWLINELemma: Let \(G\) be a critical planar graph without 6-cycles with chord. If \(\Delta(G) \geq 6\), then there is an edge \(uv \in E(G)\) such that \(\min\{ d(u), d(v)\} \leq \lfloor \frac{\Delta(G) + 1}{2} \rfloor\) and \(d(u) + d(v) \leq \max\{8, \Delta(G) + 2\}\).NEWLINENEWLINETheorem: If \(G\) is a planar graph without 6-cycles with chord, then \(\chi_{1}^{'}(G) \leq \Delta(G) + 1\) and \(\chi_{1}^{''}(G) \leq \Delta(G) + 2\), where \(\Delta(G) \geq 6\), (where \(\chi_{1}^{'}(G)\) denotes the list edge coloring number of \(G\) and \(\chi_{1}^{''}(G)\) denotes the list total coloring number of \(G\)).NEWLINENEWLINEThis gives a partial answer to the List Coloring Conjecture in the case of a planar graph \(G\) without 6-cycles with chord,NEWLINENEWLINE Conjecture: For a multigraph \(G\),NEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(a)]\(\chi_{1}^{'}(G) = \chi^{'}(G)\); NEWLINE\item[(b)]\(\chi_{1}^{''}(G) = \chi^{''}(G)\).NEWLINENEWLINENEWLINE\end{itemize}}