Odd chromatic number of graph classes

Abstract

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 proved that a connected 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. The odd chromatic number of $G$, denoted by ${\chi }_{\text{odd}}\left(G\right)$, is the minimum integer $k$ for which $G$ is $k$-odd colourable. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. We first consider a question due to Scott, which states that every graph $G$ of even order $n$ has ${\chi }_{\text{odd}}\left(G\right)\le c\sqrt{n}$, for some positive constant $c$, by proving that this is indeed the case if $G$ is restricted to having girth at least seven. We also show that any graph $G$ whose all components have even order satisfies ${\chi }_{\text{odd}}\left(G\right)\le 2\Delta -1$, where $\Delta$ is the maximum degree of $G$. Next, we show that certain interesting classes have bounded odd chromatic number. Our main results in this direction are that interval graphs, graphs of bounded modular-width all have bounded odd chromatic number. In particular, every even interval graph is 6-odd colourable, and every even graph is $3mw$-odd colourable, where $mw$ is the modular width of a graph.