### Maximal Buttonings of Non-Tree Graphs

#### Abstract

Let $G$ be a finite connected graph of $n$ vertices $v_1, v_2,\ldots, v_n$. A \emph{buttoning} of $G$ is a closed walk consisting of $n$ shortest paths $$[v_1,v_2],[v_2,v_3],\ldots,[v_{n-1},v_n],[v_n,v_1].$$The buttoning is said to be \emph{maximal} if it has a maximum length when compared with all other buttonings of $G$. The goal of this work is to find a length of a maximal buttoning of curtain graphs, such as complete multipartite graphs, grid graphs and rooted product of graphs.

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