The odd chromatic number conjecture for k-trees

Let a k-tree be a graph obtained from a complete graph on $k+1$ vertices by repeatedly adding a vertex adjacent to all vertices of an existing $k$-clique. An odd coloring of a graph is a vertex coloring in which every vertex has some color appearing an odd number of times in its neighborhood. For a positive integer $k$, let $f(k)$ be the least integer such that every $k$-tree is odd $(k+f(k))$-colorable. Odd chromatic number conjecture for k-trees. For every positive integer $k$, every $k$-tree is odd $(k+2)$-colorable. The conjecture asserts that the lower bound $f(k)\geq 2$ is attained for every $k$. It is known for $k=1,2,3$, while the general case remains open; the currently established upper bound is $k+2\left\lfloor\log_2 k\right\rfloor+3$ colors.