an:05083788
Zbl 1108.05038
Hou, Jianfeng; Liu, Guizhen; Cai, Jiansheng
List edge and list total colorings of planar graphs without 4-cycles
EN
Theor. Comput. Sci. 369, No. 1-3, 250-255 (2006).
00188605
2006
j
05C15
\textit{O. V. Borodin, A. V. Kostochka} and \textit{D. R. Woodall} [J. Comb. Theory, Ser. B 71, 184--204 (1997; Zbl 0876.05032)] proved that if \(G\) is a simple planar graph with maximum degree \(\Delta \geq 12\) then the list edge chromatic number \( \chi _{\mathrm{list}}^{\prime }(G)=\Delta \) and the list total chromatic number \(\chi _{\mathrm{list}}^{\prime \prime }(G)=\Delta +1\).
In the paper under review these equalities are shown to hold for a planar graph \(G\) which satisfies one of the following conditions: \(\Delta \geq 7\) and \(G\) has no cycle of length 4, \(\Delta =6\) and \(G\) has no cycle of length 4 or 5, or \( \Delta =5\) and \(G\) has no cycle whose length lies in the closed interval \( [4,8]\). In addition, \(\chi _{\mathrm{list}}^{\prime }(G)=\Delta \) is shown to hold for a planar graph \(G\) with \(\Delta =4\) if \(G\) has no cycle whose length lies in the closed interval \([4,14]\).
Lorenzo Traldi (Easton)
Zbl 0876.05032