A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of G. Scott [Graphs and Combinatorics, 2001] proved that a graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G is k-odd colourable if it can be partitioned into at most k odd induced subgraphs. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. In particular, we consider for a number of classes whether they have bounded odd chromatic number. Our main results are that interval graphs, graphs of bounded modular-width and graphs of bounded maximum degree all have bounded odd chromatic number.
翻译：如果每个顶点都具有奇异的(分别是、甚至)度度,图形就被称为奇数。 Gallai 证明了每个图形都可以被分割成两个甚至诱导的子图, 也可以被分割成一个甚至诱导的奇数或奇数和偶引导的子图。 我们把奇数子图的分区称为G. Scott [Graphs和Compatoricic, 2001] 的奇色。 我们主要的结果是,一个图形如果有偶数的顶点, 也只能承认一种奇数的颜色。 我们说, 图表G 如果能在最多 k 诱导的子图中被分割成两个, 就会有 kd 奇数的颜色。 我们开始系统地研究图表类别中的奇色和奇数。 特别是, 我们考虑的类别是, 它们是否包含奇数的奇数。 我们的主要结果是, 间图、 组合组合组合的图和约束最大程度的图形都含有奇异数 。</s>
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