Hanlai Lin | Searcharxiv

arXiv2026

A note on the saturation number for unions of three cliques

A graph $G$ is $F$-saturated if $G$ contains no copy of $F$ but $G+e$ contains a copy of $F$ for every missing edge $e$ of $G$. The saturation number $\sat(n,F)$ is the minimum number of edges in an $n$-vertex $F$-saturated graph. Motivated by a problem posed by Faudree, Ferrara, Gould, and Jacobson concerning $K_p\cup K_q\cup K_{q+1}$, we determine the saturation number and the unique extremal graph for $K_p\cup K_q\cup K_r$ whenever $2\le p\le q<r<p+q$ and $n$ is sufficiently large. Together with the previously known results for $r\ge p+q$ and for $r=q$, this completes the determination of the saturation number and the extremal graphs for unions of three cliques, for all sufficiently large $n$.

math.CO↗

arXiv2022

On order isomorphisms intertwining semigroups for Dirichlet forms

This paper is devoted to characterizing the so-called order isomorphisms intertwining the $L^2$-semigroups of two Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of $h$-transformation and quasi-homeomorphism. In addition, under the absolute continuity condition on Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups is the composition of $h$-transformation, quasi-homeomorphism, and multiplication by a certain step function.

math.FA↗