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

For two integers s >= 0,t >= 0, a graph G is (s,t)-supereulerian, if for every pair of disjoint subsets X,Y subset of E(G), with |X|<= s,|Y|<= t, G has a spanning eulerian subgraph H with X subset of E(H) and Y boolean AND E(H)=Phi. 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 is an element of S is (s,t)-supereulerian if and only if kappa '(G)>= 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. (c) 2024 Published by Elsevier B.V.