Odd coloring of $k$-trees

Abstract

An odd coloring of a graph is a proper coloring such that every non-isolated vertex has a color that appears at an odd number of its neighbors. This notion was introduced by Petr\v{s}evski and \v{S}krekovski in 2022. In this paper, we focus on odd coloring of $k$-trees, where a $k$-tree is a graph obtained from the complete graph of order $k+1$ by recursively adding a new vertex that is joined to a clique of order $k$ in the former graph. It follows from a result of Cranston, Lafferty, and Song in 2023 that every $k$-tree is odd $(2k+1)$-colorable. We improve this bound to show that every $k$-tree is odd $\left(k+2\left\lfloor\log_2 k\right\rfloor+3\right)$-colorable. Furthermore, when $k=2,3$, we show the tight bound that every 2-tree is odd $4$-colorable and that every 3-tree is odd $5$-colorable.