Title：On construction for trees making the equality hold in Vizing's conjecture

Abstract：In 1968, Vizing gave a conjecture:~$\gamma(G \square H)\geq\gamma(G)\gamma(H)$ for any graphs $G$ and $H$, which is now still open. In the text book ``Domination in Graph:~Advanced Topices" edited by Haynes~et~al., they listed such a question: is there a structural characterization of the graphs $G$ such that there exists a graph $H$ with $\gamma(G \square H)=\gamma(G) \gamma (H)$? In this paper, we restrict $G$ to be a tree and prove that: for any tree $T$, there exists a nonempty graph $H$ such that $\gamma(T \square H)=\gamma(T) \gamma (H)$ if and only if $\gamma(T \square C_4)=\gamma(T) \gamma (C_4)$. Based on this, we obtain a construction of the tree class $\mathscr{T}=\{\hspace{2pt}T\mid T$ is a nontrivial tree and there exists a nonempty graph $H$ such that $\gamma(T \square H)=\gamma(T)\gamma (H)\}$.

Speaker：Weisheng Zhao, Ph.D., Jianghan University. His research interests include graph theory, especially on graph colorings.

Date：3:00pm-4:00pm 2022-11-10 (Thursday).

Tencent Meeting ID：227-556-401

Organizer：School of Mathematical Science

Students and teachers who are interested in graph theory are welcome.