The Minimal Automorphism-Free Tree

A finite tree $T$ with $|V(T)| \geq 2$ is called {\it automorphism-free} if there is no non-trivial automorphism of $T$. Let $\mathcal{AFT}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism) ordered by $T_1 \preceq T_2$ if $T_1$ can be obtained from $T_2$ by successively deleting one leaf at a time in such a way that each intermediate tree is also automorphism-free. In this paper, we prove that $\mathcal{AFT}$ has a unique minimal element. This result gives an affirmative answer to the question asked by Rupinski.Comment: 8 pages, 5 figure

Tree-chromatic number is not equal to path-chromatic number

For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the chromatic number of $(T, \mathcal{B})$ is the maximum of $\chi(G[B])$, taken over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the minimum chromatic number of all tree-decompositions $(T, \mathcal{B})$ of $G$. The path-chromatic number of $G$ is defined analogously. In this paper, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are different. This settles a question of Seymour. Our results also imply that the path-chromatic numbers of the Mycielski graphs are unbounded.Comment: 11 pages, 0 figure