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：如果一个图的每个顶点的度数都是奇数（偶数），则称该图为奇图（偶图）。Gallai证明了每个图都可以划分为两个偶诱导子图，或一个奇诱导子图与一个偶诱导子图。我们将图划分为奇子图的过程称为G的奇染色。Scott[Graphs and Combinatorics, 2001]证明了一个图存在奇染色当且仅当其顶点数为偶数。若图G可被划分为至多k个奇诱导子图，则称G是k-奇可染的。我们首次系统性地研究图类的奇染色与奇色数问题。具体而言，我们考察若干图类是否具有有界奇色数。主要结论是：区间图、有界模宽图以及有界最大度图均具有有界奇色数。