The IBS discrete mathematics group welcomes Prof. Xin Zhang (张欣), a new visiting scholar from Xidian University, China. He is visiting the IBS discrete mathematics group for one year from August 22, 2019.

He received his Ph.D. from Shandong University in China and is currently an associate professor in the School of Mathematics and Statistics, Xidian University, Xi’an, China.

On May 16, 2019, Xin Zhang (张欣) from Xidian University, China discusses various results on the equitable tree-k-coloring of graphs, a problem of partitioning the vertex set of a graph into vertex-disjoint induced subgraphs having no cycles with almost same size. The title of his talk was “On equitable tree-colorings of graphs”. Xin Zhang is visiting IBS Discrete Mathematics Group from May 15 to May 19.

An equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class (i.e, the set of vertices in a common color) induces a forest and the sizes of any two color classes differ by at most one. The minimum integer $k$ such that a graph $G$ is equitably tree-$k$-colorable is the equitable vertex arboricity of $G$, denoted by $va_{eq}(G)$. A graph that is equitably tree-$k$-colorable may admits no equitable tree-$k’$-coloring for some $k’>k$. For example, the complete bipartite graph $K_{9,9}$ has an equitable tree-$2$-coloring but is not equitably tree-3-colorable. In view of this a new chromatic parameter so-called the equitable vertex arborable threshold is introduced. Precisely, it is the minimum integer $k$ such that $G$ has an equitable tree-$k’$-coloring for any integer $k’\geq k$, and is denoted by $va_{eq}^*(G)$. The concepts of the equitable vertex arboricity and the equitable vertex arborable threshold were introduced by J.-L. Wu, X. Zhang and H. Li in 2013. In 2016, X. Zhang also introduced the list analogue of the equitable tree-$k$-coloring. There are many interesting conjectures on the equitable (list) tree-colorings, one of which, for example, conjectures that every graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for any integer $k\geq (\Delta+1)/2$, i.e, $va_{eq}^*(G)\leq \lceil(\Delta+1)/2\rceil$. In this talk, I review the recent progresses on the studies of the equitable tree-colorings from theoretical results to practical algorithms, and also share some interesting problems for further research.