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List edge and list total colorings of planar graphs without 4-cycles. (English) Zbl 1108.05038
O. V. Borodin, A. V. Kostochka and 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 \([4,14]\).

05C15 Coloring of graphs and hypergraphs
Full Text: DOI
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