Structural properties and edge choosability of planar graphs without 6-cycles (Q2731587)

The list chromatic index of a graph \(G\) is denoted by \(\chi'_l(G)\). A well-known conjecture of Vizing states that for every graph \(G\) with maximum degree \(\Delta(G)\), \(\chi'_l(G) \leq \Delta(G) + 1\). This conjecture has been proved for graphs with maximum degree 3 by Harris [PhD dissertation, Cambridge University, UK, 1984] and for graphs with maximum degree 4 by \textit{M. Juvan, B. Mohar} and \textit{R. Škrekovski} [J. Graph Theory 32, No. 3, 250-264 (1999; Zbl 0934.05052)]. The present authors prove that for planar graphs \(G\) without 6-cycles, \(\chi'_l(G) \leq \max \{7, \Delta(G)+1 \}\). This and the results of Harris and Juvan et al. imply that the conjecture of Vizing is true for planar graphs \(G\) without 6-cycles and \(\Delta(G) \not= 5\).