Postman tours and cycle covers (Q686511)

Let \(G= (V,E)\) be a graph. It is shown that if \(G\) is bridge-less then \(G\) has a postman tour of length at most \(| E(G)|+ | V(G)|- 3\) and if \(G\) is minimally 2-edge-connected then \(G\) has a postman tour of length at most \(2| V(G)|- 2\). With the aid of a theorem of \textit{Alspach}, \textit{Goddyn} and the reviewer [Trans. Am. Math. Soc. (to appear)], the graph \(G\) has a shortest circuit cover with the above total length if \(G\) contains no subdivision of the Petersen graph.