Squares of graphs are optimally (s,t)-supereulerian

Abstract

For two integers $s\ge 0,t\ge 0$, a graph $G$ is $(s,t)$-supereulerian, if for every pair of disjoint subsets $X,Y\subset E\left(G\right)$, with $\left|X\right|\le s,\left|Y\right|\le t$, $G$ has a spanning eulerian subgraph $H$ with $X\subset E\left(H\right)$ and $Y\cap E\left(H\right)=0̸$. Pulleyblank (1979) proved that even within planar graphs, determining if a graph $G$ is $(0,0)$-supereulerian is NP-complete. Xiong et al. (2021) identified a function $j_0(s,t)$ such that every $(s,t)$-supereulerian graph must have edge connectivity at least $j_0(s,t)$. Examples have been found that having edge connectivity at least $j_0(s,t)$ is not sufficient to warrant the graph to be $(s,t)$-supereulerian. A graph family $S$ is optimally $(s,t)$-supereulerian if for every pair of given non-negative integers $(s,t)$, a graph $G\in S$ is $(s,t)$-supereulerian if and only if ${\kappa }^{\prime }\left(G\right)\ge j_0(s,t)$. Hence the $(s,t)$-supereulerian problem in such a graph family can be solved in polynomial time with minimally required edge-connectivity. In this research, we prove that the family of all squares of graphs of order at least 5 is optimally $(s,t)$-supereulerian.