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...gs of planar graphs with large girth 345 Fact 1 (obvious). of its vertices. If χ a (G) � k then G contains an induced forest on at least 2�k Fact 2 (S. L. Hakimi, J. Mitchem and E. S. Schmeichel (see =-=[6]-=-)). If χ (G) � k a then E(G) can be partitioned into k ‘star forests’ (forests in which each component is a star). Fact 3 (Gru� nbaum [5]). k�2k−�. If χ a (G) � k then the star chromatic number χ s (G...

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...graph G, Gru� nbaum [5] conjectured that χ (G) � 5 and proved that a χ (G) � 9. This bound was sharpened by Mitchem [9] to 8, by Albertson and Berman a [1] to 7, by Kostochka [7] to 6, and by Borodin =-=[3, 4]-=- to 5, which is best possible since the double 5-wheel C �K� is planar and (it is easy to see) has χ � 5. � � a The girth g � g(G) of a graph G is the length of its shortest cycle. The purpose of the ...

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...(G) � Fact 4 (Raspaud and Sopena [10]). number χ (G) � k�2k−�. o If χ a (G) � k then the oriented chromatic By Fact 2, Borodin’s 5-colour theorem implies the truth of the conjecture of Algor and Alon =-=[2]-=- that the edges of every planar graph can be partitioned into five star forests. By Facts 3 and 4, it also implies that χ (G) � 80 and χ (G) � 80 for every s o planar graph G; these bounds remain the ...

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...n be partitioned into k ‘star forests’ (forests in which each component is a star). Fact 3 (Gru� nbaum [5]). k�2k−�. If χ a (G) � k then the star chromatic number χ s (G) � Fact 4 (Raspaud and Sopena =-=[10]-=-). number χ (G) � k�2k−�. o If χ a (G) � k then the oriented chromatic By Fact 2, Borodin’s 5-colour theorem implies the truth of the conjecture of Algor and Alon [2] that the edges of every planar gr...

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...rtly inspired by J. Nes� etr� il telling us of Fact 4 (below). Theorem 1. If G is planar with girth g � 5 then χ a � 4. Theorem 2. If G is planar with girth g � 7 then χ a � 3. Kostochka and Melnikov =-=[8]-=- have constructed planar 2-degenerate bipartite graphs, necessarily with girth g � 4, having χ a � 5. (For example, in C � �K� � , replace each edge u� of C � by a copy of K �,� with u, � as the verti...

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...graph G, Gru� nbaum [5] conjectured that χ (G) � 5 and proved that a χ (G) � 9. This bound was sharpened by Mitchem [9] to 8, by Albertson and Berman a [1] to 7, by Kostochka [7] to 6, and by Borodin =-=[3, 4]-=- to 5, which is best possible since the double 5-wheel C �K� is planar and (it is easy to see) has χ � 5. � � a The girth g � g(G) of a graph G is the length of its shortest cycle. The purpose of the ...

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...le and χ (F) � 2ifF is a forest, with equality unless F is edgeless. a For a planar graph G, Gru� nbaum [5] conjectured that χ (G) � 5 and proved that a χ (G) � 9. This bound was sharpened by Mitchem =-=[9]-=- to 8, by Albertson and Berman a [1] to 7, by Kostochka [7] to 6, and by Borodin [3, 4] to 5, which is best possible since the double 5-wheel C �K� is planar and (it is easy to see) has χ � 5. � � a T...

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...h equality unless F is edgeless. a For a planar graph G, Gru� nbaum [5] conjectured that χ (G) � 5 and proved that a χ (G) � 9. This bound was sharpened by Mitchem [9] to 8, by Albertson and Berman a =-=[1]-=- to 7, by Kostochka [7] to 6, and by Borodin [3, 4] to 5, which is best possible since the double 5-wheel C �K� is planar and (it is easy to see) has χ � 5. � � a The girth g � g(G) of a graph G is th...

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...edgeless. a For a planar graph G, Gru� nbaum [5] conjectured that χ (G) � 5 and proved that a χ (G) � 9. This bound was sharpened by Mitchem [9] to 8, by Albertson and Berman a [1] to 7, by Kostochka =-=[7]-=- to 6, and by Borodin [3, 4] to 5, which is best possible since the double 5-wheel C �K� is planar and (it is easy to see) has χ � 5. � � a The girth g � g(G) of a graph G is the length of its shortes...

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... a a the smallest number of colours in an acyclic colouring of G. Clearly χ (C) � 3ifCis a a cycle and χ (F) � 2ifF is a forest, with equality unless F is edgeless. a For a planar graph G, Gru� nbaum =-=[5]-=- conjectured that χ (G) � 5 and proved that a χ (G) � 9. This bound was sharpened by Mitchem [9] to 8, by Albertson and Berman a [1] to 7, by Kostochka [7] to 6, and by Borodin [3, 4] to 5, which is b...